Ancient Greek religion consisted mainly of the worship of the twelve Olympian gods, as well as participation in Panhellenic festivals such as the Olympic games, and recognition of sites of religious importance such as the Oracle of Delphi. There was no dogma, or set of beliefs to which the Greeks had to subscribe to, and there was no official interpretation of religious observance.

Greek religion was so intertwined with their society and daily lives that, interestingly, there was no Greek word for 'religion'. Religious practices consisted of various actions, such as festivals, sacrifices and everyday customs such as purification rituals. At the heart of Greek religion were religious cults, which varied both within individual poleis (city states) and across Greece. Different cults attributed different epithets to the Olympian gods to reflect their own local culture, for example, Athena was worshipped under around fifty epithets such as Athena Parthenos (Virgin) or Athena Boulaia (Counsellor). It can be argued that these epithets created completely different gods to those worshipped by other cults.

There were no canonical religious documents in ancient Greece, however, tales by Homer and Hesiod tell stories about the actions of the gods, as well as the heroic deeds of humans; these stories were told by travelling bards such as Homer. Stories such as Homer's Odyssey and Iliad were probably composed from older tales which were widely told across Greece by travelling bards. Heroes were also important to Greek religion, as hero cults such as the cult of Asclepius, the Greek hero and god of medicine, worshipped heroic figures.

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For further reading on Greek religion, I would recommend Religion and the Greeks by Robert Garland.

I can provide sources if anyone ask me for them. Here, however, is what I remember based upon reading about Greek myths and religions and studying them briefly in university. All religions can, perhaps, be divided into several parts that mix together in various ways: the ritual, the belief, the myth, and the meaning assigned to the myth. From this prism, Greek myths may not be a sound basis for a religion, but they are only one portion of the wider Greek religious experience in which myth could and did form an integral part.

Take, for example, the myth about how Demeter’s daughter Persephone was abducted by Hades and forced to marry him, following which Demeter wandered the world in search of her daughter, stayed at Eleusis where she befriended its ruling family, and eventually gained some months of the year back with her daughter.

The meaning assigned to the myth at Eleusis (a city near Athens) however, made the story much more significant than the brief outline that I have provided. Demeter, the goddess of fertility, was said to have taught the ruling family of Eleusis certain truths about how to ensure the best fate for themselves after their deaths. During the Classical Period and later, there were two dynasties of priests at Eleusis, the Eumolpidae and the Kerykes, who claimed descent from members of Eleusis’s ruling family who had learned these truths from Demeter when she had been at Eleusis. Each clan’s leader bore the title Hierophant (meaning “to show the holy”), which refers to their roles as instructors of people who sought assurances that they would, after death, have a good fate.

The ritual performed at Eleusis, presided over by the Hierophants, was secretive, but it involved certain predicable activities, whose performance was apparently alluded to (and justified for the more cynical modern reader) by certain details in the full cycle of stories about Demeter’s stay in Eleusis and meeting with various people.

The precise details of the ritual and what it taught are to this day unknown, although theories exist. Regardless, there are records of pious people who, having gone through the rituals and teachings at Eleusis, believed that they would be guaranteed a fine fate after their deaths.

All of these features – ritual, secret teachings, belief by initiates into the mysteries, grew out of the myth and had the myth at their core, but without such accoutrements the myth of Demeter’s search for Persephone may seem to be not a sound basis for a religion. With these accoutrements, however, it was the basis for a religion that lasted for hundreds of years, until the Arian Christian Goths, in anti-Pagan zeal, sacked Eleusis during the late 4th century CE.