It is my understanding that spectators at Athenian dramatic festivals in the 5th c. BCE had a detailed familiarity with the myths that were adapted by Greek dramatists, and that the dramatists in turn, aware of this prior knowledge, played upon audience expectations (e.g. Sophocles in his Electra leading the audience at one point to think that he will alter the myth and have Clytemnestra be murdered by her daughter rather than her son).
Where did this familiarity come from? What do we know about how stories about the gods circulated in classical Athens, other than through dramatic adaptations? Did many Athenians have access to written texts of Homer and Hesiod? Did stories of the gods circulate mainly through oral culture, like folk tales, passed on from parents to children and so on? Were there religious festivals--in addition to the dramatic ones--at which such stories were recounted and thus became familiar?
It is actually very debatable how much the average audience member would know a given story before watching a play. For example, in Aristotle's Poetics when he is discussing how to use "real" stories he says:
We must not, therefore, at all costs keep to the received legends, which are the usual subjects of Tragedy. Indeed, it would be absurd to attempt it; for even subjects that are known are known only to a few, and yet give pleasure to all.
This strongly implies that the sort of detailed knowledge found in a modern commentary was only known to a small portion of the audience. It is pretty unlikely that anyone watching, say, Agamemnon wouldn't know how that ends up (really, the dramatic tension depends on that), but what about someone watching Antigone or Alcestis? And of course, not everyone's internal story would be the same. Each myth had innumerable variations large and small, and so two people who "knew" a myth would have completely different versions. It is also quite common for mythographers an storytellers (Pindar has a well known example) that X is one story that people say, but in reality it happened like Y--which emphasizes how many different channels of knowledge there were.
For the actual transmission of knowledge, the oral and festival methods you mention seem quite probable, although unfortunately we cannot do the sort of anthropological fieldwork neccesary to reconstruct the ordinary Greek systems of knowledge. One particular method suggested by scholars like Oliver Taplin is pilgrimage, as Greeks from diverse areas would converge on single spots and create environments where knowledge can be transferred and standardized. We can fairly certain that it was largely not texts, however, which were generally not compiled until the Hellenistic and were not accessible to much of the population even then.