Then, in place of all other pleasures, introduce that of being conscious that you’re obeying God, and that you’re accomplishing, not in mere word but in very deed, the work of a good and virtuous person.

Epictetus, Discourses 3.24.110

Accustom yourself to be master of the following:
first of all, of your stomach, of sleep, of lust,
and of anger.

The Golden Verses 9-11

A very quick guide to getting started:

1. On a regular basis, wash your hands, make an offering, and pray. While washing your hands, ask for purity of body and mind. If you don’t currently have incense, wine, food, or other such things to offer, offer a hymn.
2. If you have no idea what to pray about, pray for understanding of the goodness of the Gods and that they may give to you what they know to be best.
3. Read and study the Golden Verses (they’re quite short), a chapter or two of The Handbook by Epictetus, or some other appropriate text. Both are available for free online.
4. Every night before you sleep, examine yourself and your day. Acknowledge as cold fact everything you did and didn’t do, the good and the bad. It is not about guilt, it is about becoming Godlike -- but that requires radical self-honesty.

You will learn the rest as you go.

"...and that when the official and his company had poured libations to whatever gods that day and night happen to be dedicated..." Plato, Laws 807a

One simple way of choosing a God to dedicate your offering or prayer to is by the day of the week. In many modern European languages, the days of the week are named after divinities. As you probably know, the Germanic deities are (mostly) used in the English days, and these correspond to those used by the Greeks and Romans:

Monday - Selene / Luna
Tuesday - Ares / Mars
Wednesday - Hermes / Mercury
Thursday - Zeus / Jupiter
Friday - Aphrodite / Venus
Saturday - Kronos / Saturn
Sunday - Helios / Sol

One way to learn about the Gods is to read (or sing) a hymn or myth about them on these days. Of course, that's an imperfect method since there are many Gods besides these seven, but it's a start. Calendars can be consulted for additional information. (Having an index to Plato's dialogues is handy, because it allows you to find passages where specific Gods are discussed.) The more you learn about the Gods, the more keys you will find to unlocking their proper worship.

The time of day can be taken into account as well. The most auspicious time to honor the Olympians is in the morning (though it's appropriate to worship them at any time); heroes, in the afternoon; and the Underworld Gods and ancestors, in the evening. (See Diogenes Laertius The Life of Pythagoras 19, William Guthrie The Greeks and Their Gods p 222.).

When offering to Olympians, it’s best to use your right hand and offer in an odd number. When offering to Underworld Gods or ancestors, it’s best to use your left hand and offer in an even number. (See Plato Laws 717b and the Pythagorean akousma: “To the celestial Gods sacrifice an odd number, but to the infernal, an even.” It's also worth noting that, to this day, it is the very interesting custom in parts of Eastern Europe to offer gifts - of flowers, for example - to the living in an odd number but to the dead in an even number.)

I hope this information helps you.

As it is, to dedicate your life to winning a victory at Delphi or Olympia keeps you far too busy to attend to other tasks; but a life devoted to the cultivation of every physical perfection and every moral virtue (the only life worth the name) will keep you at least twice as busy. Inessential business must never stop you taking proper food and exercise, or hinder your mental and moral training. To follow this regimen and to get the maximum benefit from it, the whole day and the whole night is scarcely time enough.

Plato, Laws 807c-d

Most honored of immortals, many-named one, ever omnipotent,
Zeus, prime mover of nature, steering all things by your law,
Hail!
For it is proper for all mortals to speak to you:
For we all descend from you, bearing our share of your likeness
We alone, of all mortal creatures that live and move on earth.
So, I shall make song of you constantly and sing forever of your might.
Truly, this whole universe, spinning around the earth,
Obeys you wherever you lead, and willingly submits to your rule;
Such is the servant you hold in your unconquerable hands,
A double-edged, fiery, ever-living thunderbolt.
For by its strikes all the works of nature happen.
By it you direct the universal reason, which pervades all things
Intermixing with the great and small lights of the heavens.
Because of this you are the greatest, the highest ruler of all.
Not a single thing that is done on earth happens without you, God,
Nor in the divine heavenly sphere nor in the sea,
Except for what bad people do in their foolishness.
But you know how to make the crooked straight
And to bring order to the disorderly; even the unloved is loved by you.
For you have so joined all things into one, the good and the bad,
That they all share in a single unified everlasting reason.

It is shirked and avoided by all the wicked among mortals,
The wretched, who ever long for the getting of good things,
Neither see nor hear God’s universal law,
By which, obeying with understanding, they could share in the good life.
But instead they chase after this and that, far from the good,
Some in their aggressive zeal for fame,
Others with a disordered obsession with profits,
Still others in indulgence and the pleasurable exertions of the body.
[They desire the good] but are carried off here and there,
All the while in zealous pursuit of completely different outcomes.
But bountiful Zeus, shrouded in dark clouds and ruling the thunder,
Protect human beings from their ruinous ignorance;
Scatter it from our souls, grant that we might obtain
True judgment on which you rely to steer all things with justice;

So that having won honor, we may honor you in return,
Constantly singing of your works, as it is proper
For mortals to do. For neither mortals nor gods have any greater privilege
Than to make everlasting song of the universal law in justice.

Cleanthes, Hymn to Zeus

One must sacrifice to the gods for three purposes: because of honour, because of charis, or because of one’s need for good things. For just as we think we must make first-fruit offerings to good men, so we think we must make them also to the gods. We honour the gods when we are seeking that there be for us either a turning away of evils or the preparation of good things, or after we have had good experiences and not for the purpose of obtaining some (additional) benefit, or in the simple honouring of their good disposition toward us.

Theophrastus, On Piety fragment 12.42–8

Though [Socrates'] sacrifices were humble, according to his means, he thought himself not a whit inferior to those who made frequent and magnificent sacrifices out of great possessions. The gods (he said) could not well delight more in great offerings than in small — for in that case must the gifts of the wicked often have found more favour in their sight than the gifts of the upright — and man would not find life worth having, if the gifts of the wicked were received with more favour by the gods than the gifts of the upright. No, the greater the piety of the giver, the greater (he thought) was the delight of the gods in the gift. He would quote with approval the line:

According to thy power render sacrifice to the immortal gods,

and he would add that in our treatment of friends and strangers, and in all our behaviour, it is a noble principle to render according to our power.

Xenophon, Memorabilia 1.3.3

"He was more careful in observing the unlucky days of the Egyptians than they are themselves, and maintained a personal fast on certain days through revelation. For he went without food on every 'old and new,' not even having dined the previous day, just as indeed he also performed conspicuous and holy rituals at the new moons." - Marinus, Proclus, or on Happiness 19 (trans. Mark Edwards)

It seems the Greek here is somewhat ambiguous, but according to the translations of Thomas Taylor and Mark Edwards, Proclus fasted on the last day of the month (Guthrie's translation suggests it was on the first). According to Edwards, 'old and new' was the Athenian term for the last day of the lunar month and does not mean "last and first days of the months." Marinus does not provide detail about the rituals Proclus performed on the new moon, but there are various statements in the literature about new moon practices, such as libations to the moon (see note 206 in Edwards).

The next new moon is on the thirtieth of May.

"By paying honor and respect to one’s kinfolk and all who share in the worship of the gods of the tribe and who also share descent and blood, a person will also enjoy the favor of the gods of the household who will be well disposed toward his own begetting of children."

Plato, Laws 729c

[The daimonic] interprets and conveys things from humans to gods and things from gods to humans, that is, the requests and sacrifices of the humans and the orders and returns for sacrifices of the gods. Being between both it fills (the void) so that everything is bound together. Through the daimonic all divination proceeds and the craft of the priests who are concerned with sacrifices, rituals, chants, and all divination and sorcery. God does not mix with human, but through the daimonic is all association and discussion for gods with humans, both when humans are awake and asleep ... These daimones are numerous and of all kinds, and Eros is one of these.

Plato, Symposium 202e-203a

You'll have to excuse the length of this post, but I thought it was worth sharing since it is one of the more detailed denoscriptions of ancient sacrifice in Europe.

"[The ox] is sacrificed if it makes a deep bow to Hestia. The kings' share-receiver makes the sacrifice and provides the offerings and offers in addition a half hekteus of offerings. He takes his share, the skin and a leg, the hieropoioi take a leg, and the remaining meat belongs to the city.

The heralds lead the ox selected for Zeus to the agora. After reaching the agora, the owner of the ox or the person deputizing for him calls out: 'For the people of Cos I am providing the ox; let the Coans give the price to Hestia.' And let the chief magistrates immediately take an oath and make a valuation; when a valuation has been made, let the herald announce how much the valuation was. Then they drive the ox to Hestia Hetaeraea and make the sacrifice. The priest puts a ribbon upon the ox and pours a cup of wine mixed with water as a libation in front of the ox. Then they lead the ox away, together with the burnt offering, seven cakes, honey, and the ribbon. As they leave, they call for respectful silence. At this point they untie the ox and begin the sacrificial ritual with olive and laurel. The heralds burn a pig and its entrails upon the altar, pouring on libations of honey and milk. After washing the intestines, they burn them beside the altar. And once they are burnt without wine, let him pour upon them a libation of honey and milk.

Let the herald announce that they are celebrating the annual festival as a feast for Zeus Polieus. Let the priest make an additional offering together with the intestines, incense, and cakes, libations mixed and unmixed, and a ribbon. Then let the priest and herald go to the hieropoioi at the public building, and let the hieropoioi entertain the priest and the herald during this night. And let the heralds choose whoever they want from their own number as slaughterer of the ox and let whoever of them wishes to proclaim the choice to the person chosen.

On the same day: to Dionysius Scyllites, a pig and a kid. The pig meat is not to be taken away [from the sanctuary]. The priest makes the sacrifice and provides the offerings. As perquisites, he takes the skin and leg.

On the twentieth: the selected ox is sacrificed to Zeus Polieus. What has to be wrapped is wrapped in the skin. On the hearth a sacrifice is made of half a hekteus of barley, two half-hekteus loaves, one shaped like a cheese, and things wrapped in skin. On these the priest pours a libation of three mixing bowls of wine. Perquisites from the ox: for the priest the skin and a leg (the priest provides the offerings) and half of the breast and half the stomach; for the incense-bearer the hip-end of the leg given to the hieropoioi; for the heralds, a double portion of meat from the back, shoulder meat, a three-spit share of blood meat; for the Nestoridae, a double portion of meat from the back; for the doctors, meat; for the flute-player, meat; to the smiths and potters, the brain. The rest of the meat is for the city. All of these are not taken out of the city.

On the same day: to Athena Polias, a pregnant sheep. The priest makes the sacrifice and provides the offerings. As perquisites, he takes the skin and a leg. On the twenty-first: to Dionysius Scyllites a pig and a kid: the meat of the pig is not to be taken away. The priest makes the sacrifice and provides the offerings. As perquisites he takes the skin and a leg."

A denoscription of sacrifice in Cos to Zeus Polieus (of the City) from an innoscription on a fourth century BCE calendar. Quoted from Greek Religion: A Sourcebook by Valerie Warrior p. 67.

As we train ourselves to deal with sophistical questioning, so we should also train ourselves each day to deal with impressions, because they too put questions to us.

'The son of So-and-so has died.' - Reply: That lies outside the sphere of choice, it is nothing bad.
'So-and-so has been disinherited by his father.' - That lies outside the sphere of choice, it is nothing bad.
'Caesar has condemned him' - That lies outside the sphere of choice, it is nothing bad.
'He has been distressed by these things.' - That lies within the sphere of choice, it is something bad.
'He has endured it nobly.' - That lies within the sphere of choice, it is something good.

If we adopt this habit, we'll make progress, because we'll never give our assent to anything unless we get a convincing impression.

'His son has died.' - What has happened? - 'His son has died.' - Nothing more than that? - 'Nothing more.'
'His ship has gone down.' - What has happened? His ship has gone down.
'He has been taken off to prison.' - What has happened? He has been taken off to prison. But the observation 'Things have gone badly for him' is something that each person adds for himself.

Epictetus, Discourses 3.8

As for the famous judgment of the goddesses that the myths say was performed by Paris, following the ancient account, it is not to be believed that there was truly strife among the goddesses themselves and that they were judged by a [particular] barbarian. Rather, this is to be interpreted as meaning that the choices of lives – to which Plato testifies in many passages – are likewise carried out under the watchful eye of the gods who supervise souls.

Plato himself indeed clearly teaches the same thing in the Phaedrus, saying that the regal life belongs to Hera, the philosophical to Zeus, and the erotic to Aphrodite. Thus souls, when many kinds of lives are offered them out of the universe, accept some and reject others, following their own judgment, while the myths, transferring to the gods themselves the specific qualities of the lives, say that those who preside over the variation in them, form by form, are "judged" by those choosing the lives.

This is the sense in which Paris is said to have been made the judge of Athena, Hera, and Aphrodite: three lives were offered him, and he chose the erotic, not after due thought, but rushing after the beauty of the world of appearances and pursuing the phantom of the beauty grasped by the mind. He whose life is truly devoted to Eros sets intelligence and wisdom before him and contemplates the true and the apparent beauty through these and has no less to do with Athena than with Aphrodite. But he who pursues only the erotic form of life, in and for itself and through the passions, departs from true beauty and goodness and out of stupidity and greed leaps upon the phantom of the beautiful and lies there on it, failing to attain that balanced perfection commensurate with the erotic. The truly erotic individual, who is the concern of Aphrodite, is drawn up to the divine beauty itself, looking beyond the beauties of the senses, but since there are Aphrodisian daemons presiding over the beauty that is visible and has its existence in matter, for this reason, of course, even he who pursues the phantom is said to have Aphrodite as his helper.

- Proclus, On the Republic 6

'It is fitting' for each of us to do these things 'according to the customs of one's country'. For God is always simultaneously present everywhere, with all of his divine powers. But we are limited to one form among those many forms produced by God, the human form, and within the human form are limited to one form of life for now and one choice of life, and are divided up into a little portion of the universe and of the earth itself. So different people partake in a different instance of divine goodness, and they do so in a different way at different times and places. You can at least see that when it is day with us, it is night for others, and when it is winter in one place, it is summer in another, and that these sorts of flora and fauna prevail here, and elsewhere other sorts: the earth and the things on it partake of divine goodness in a divided way.

Simplicius, On Epictetus' Handbook 94.8-21

The most important thing is not life, but the good life.

Plato, Crito 48b

But all things desire the Good, and each attains it through the mediation of its own proximate cause: therefore each has appetition of its own cause also. Through that which gives it being it attains its well-being; the source of its well-being is the primary object of its appetites; and the primary object of its appetite is that upon which it reverts.

Proclus, The Elements of Theology 31

This is just what I was getting at when I said I knew of a way to put into effect this law of ours which permits the sexual act only for its natural purpose, procreation, and forbids not only homosexual relations, in which the human race is deliberately murdered, but also the sowing of seeds on rocks and stone, where it will never take root and mature into a new individual; and we should also have to keep away from any female 'soil' in which we'd be sorry to have the seed develop.

Plato, Laws 838e-839a

Likewise I advise those who cherish the ways of freedom and shun the yoke of slavery as something evil, to beware lest by an excessive and ill-timed thirst for freedom they fall into the affliction of their ancestors, the excessive anarchy they experienced as a result of their unmeasured passion for liberty. For the Sicilians before the reign of Dionysius and Hipparinus lived happily, as they thought, faring sumptuously and ruling their rulers; they it was who, without any legal judgment, stoned to death the ten generals who preceded Dionysius, in order not to be subject to any master, not even justice and the law, but to be altogether and absolutely free. This is why tyranny came upon them. Both servitude in excess and liberty in excess are very great evils, but in due measure both are great goods. Due measure is found in obedience to God, the absence of measure in obedience to men. And the god of wise men is the law; of foolish men, pleasure.

Plato(?), Letter VIII 354d-e

But souls that live according to virtue shall, in other respects, be happy; and when separated from the irrational nature, and purified from all body, shall be conjoined with the gods, and govern the whole world, together with the deities by whom it was produced.

Sallust, On the Gods and the World Chapter 21

On the day after the conference it is prudent to honour the anniversary of the death of both the divine emperor Julian and the Inspired Gemistus Plethon. May the high gods accept them and may we share in their wisdom and courage