He (the Archon king) supervises processions, the one celebrated in honor of Asclepius when initiates
keep a watch-night, and the one at the Great Dionysia, in which he acts jointly with the Supervisors;
these were formerly ten men elected by show of hands by the People, and they found the expenses of
the procession out of their own pockets, but now they are elected by lot, one from each tribe, and
given 100 minae for equipment;
[5]   and he also supervises the procession of Thargelia, and the one in honor of Zeus the Savior.

Pausanias 7.18.10-12

[11] Every year too the people of Patrae celebrate the festival Laphria in honor of their Artemis, and at
it they employ a method of sacrifice peculiar to the place. Round the altar in a circle they set up logs of
wood still green, each of them sixteen cubits long. On the altar within the circle is placed the driest of
their wood. Just before the time of the festival they construct a smooth ascent to the altar, piling earth
upon the altar steps.
[12] The festival begins with a most splendid procession in honor of Artemis, and the maiden
officiating as priestess rides last in the procession upon a car yoked to deer. It is, however, not till the
next day that the sacrifice is offered, and the festival is not only a state function but also quite a popular
general holiday. For the people throw alive upon the altar edible birds and every kind of victim as well;
there are wild boars, deer and gazelles; some bring wolf-cubs or bear-cubs, others the full-grown
beasts.

prayers:  Penelope prays to Athena: Odyssey 4.759-767

 So she spoke, and lulled Penelope's laments, and made her eyes to cease from weeping. She then bathed, and took clean raiment for her body, [760] and went up to her upper chamber with her handmaids, and placing barley grains in a basket prayed to Athena:
    ìHear me, child of Zeus who bears the aegis, unwearied one. If ever Odysseus, of many wiles, burnt to thee in his halls fat thigh-pieces of heifer or ewe, [765] remember these things now, I pray thee, and save my dear son, and ward off from him the wooers in their evil insolence.î
    So saying she raised the sacred cry, and the goddess heard her prayer.

sacrifice:  Pausanias, Description of Greece V.15. 10-12

 [10] Each month the Eleans sacrifice once on all the altars I have enumerated.  They sacrifice in an ancient manner; for they burn on the altars incense with wheat which has been kneaded with honey, placing also on the altars twigs of olive, and using wine for a libation.  Only to the Nymphs and the Mistresses are they not wont to pour wine in libation, nor do they pour it on the altar common to all the gods.  The care of the sacrifices is given to a priest, holding office for one month, to soothsayers and libation-bearers, and also to a guide, a flute-player and the woodman.
[11] The traditional words spoken by them in the Town Hall at the libations, and the hymns which they sing, it were not right for me to introduce into my narrative.  They pour libations, not only to the Greek gods, but also to the god in Libya, to Hera Ammonia and to Parammon, which is a surname of Hermes.  From very early times it is plain that they used the oracle in Libya, and in the temple of Ammon are altars which the Eleans dedicated.

Rites of cleansing or purification: Odyssey 23.45. An old servant in the palace of Odysseus describes to his wife, Penelope, how her master killed the suitors who had harassed her during his long absence, and how afterwards he purified the hall:

 [45] Then I found Odysseus standing among the bodies of the slain, and they, stretched all around him on the hard floor, lay one upon the other; the sight would have warmed thy heart with cheer.  And now the bodies are all gathered together at the gates of the court, [50] but he is purging the fair house with sulphur, and has kindled a great fire, and sent me forth to call thee. Nay, come with me, that the hearts of you two may enter into joy, for you have suffered many woes. But now at length has this thy long desire been fulfilled: [55] he has come himself, alive to his own hearth, and he has found both thee and his son in the halls; while as for those, even the wooers, who wrought him evil, on them has he taken vengeance one and all in his house.

Offerings:  Iliad 4.1.44.  Zeus (quarrelling with Hera) speaks about the frequent offerings the men of Troy (= Ilion) have made to him. As a consequence he has promised to protect them and prevent the Greeks from taking the city:

For of all cities beneath sun and starry heaven [45] wherein men that dwell upon the face of the earth have their abodes, of these sacred Ilios was most honoured of my heart, and Priam and the people of Priam, with goodly spear of ash.  For never at any time was mine altar in lack of the equal feast, the drink-offering, and the savour of burnt-offering, even the worship that is our due.

Lectisternium    A public ritual banquet offered by the Romans to the Greek gods in 217 BCE. This is how the historian Livy describes it:  Livy, History of Rome, 5.13.4-5.

In 22.10.9 he refers to the lectisternium  of 217 BCE, which marks the first appearance of the twelve Olympian gods in Roman religion and the final fusion of the native and Greek gods:

A lectisternium was then celebrated during three days under the supervision of the decemvirs who had charge of sacrifices. Six couches were displayed: one for Jupiter and Juno, a second for Neptune and Minerva, a third for Mars and Venus, a fourth for Apollo and Diana, a fifth for Vulcan and Vesta, a sixth for Mercury and Ceres.