I’ve briefly touched on a few aspects of the Greek pantheon at the end of this answer (LINK), but I’ll expand a bit here.

Pre-Christian religion, especially pagan religion, doesn’t have the same structure as Abrahamic religions do; when the gods are all considered to be forces of nature at work, personified as named deities (Zeus, Athena, Hera, Demeter, etc.), you don’t have sermons as such preaching the virtues of every deity. City-states had patron deities (as /u/LegateHiberno mentioned, Athens’ patron goddess was Athena, and Lemnos’ big cult was centered around Hephaistos, and so on), but the only real hardcore worship centers were mystery cults--again, such as the Eleusinian mysteries, a ritual complex located at Eleusis which performed rituals in honor of the two goddesses Demeter and Persephone. There were periodic festivals and sacrifices dedicated to specific deities, and sacrifices were carried out for personal protection as well.

How much do you know about the mythology as a citizen? First off, myth and culture were inextricably intertwined in ancient Greece, and most definitely in Athens. You have what’s generally referred to as mytho-history, which, among other things, encompasses Homer’s Iliad and Odyssey, which describe human-divine interaction and its influence on the course of Greek history. This mytho-history also extends to the histories of the city-states themselves; for Athens in particular, Theseus was co-opted from mytho-history at large as the “identifying hero” of Athens, the former king.

As for the “average Greek,” (which I here define as “native-born Athenian citizen”) yes, you would probably have known your history and your Homer, and may have been looked at a bit sideways for not knowing it; I base this argument on the plays written by Aristophanes for the city Dionysia (drama competition sacred to Dionysus), which was called the Lenaia, and on plays such as Seven against Thebes and Oedipus at Colonus. Aristophanes makes jokes about Zeus’s exploits pretty frequently, and makes reference to Heracles as a voracious character, as well as including Dionysus and Hades as characters in his play Frogs. The other two plays I mentioned, Aeschylus’ Seven against Thebes and Sophocles' Oedipus at Colonus, detail specific mythical events in the history of Thebes, but were written by Athenians, broadly speaking (Aeschylus fought for Athens, and Sophocles worked as a proboulos when the Sicilian expedition failed). The myths were not generated by the playwrights; the tragic rules dictated that tragedies must either depict established historical events or known mythical episodes, and both playwrights expected their audiences to know the myth of Oedipus and the house of Laius beforehand.

This is also the case with plays like Ajax or Agamemnon, which both portray events from Homeric tradition; Ajax even includes both Odysseus and Athena as characters. Heracles shows up in Sophocles’ Philoctetes; the mythical backdrop to Greek history is very present in both dramatic and plastic art from the fifth century.

Would you have taken it all as historical fact? That’s a complicated question, simply by nature of our definitions of “history” and “fact.” These myths are how the identities of the city-states were established, and how families traced back their lineage. I talked about Theseus earlier; let me use him as an example.

In Oedipus at Colonus, Theseus is treated as an embodiment of Athens, as is the case for basically any tragedy in which he features as a character (another example is Euripides’ Suppliants, which portrays Theseus differently, but still treats him as the embodiment of Athenian character—for better or worse). Backtracking a bit, the first Athenian king was a guy called Kekrops, allegedly born from the earth and who was half-human, half-serpent. Eventually, the Athenian royal lineage took on human form in Theseus, who was, according to Buxton, “born at a time when the Athenian royal succession was in a state of uncertainty, being divided between four brothers”—another aspect of Athenian mytho-history. After “qualifying” himself with exploits such as the defeat of the minotaur, Theseus became the king of Athens, and “a quintessentially political figure,” combining the small villages of Attica into a cohesive state with Athens as its cultural and political center. The myth of Theseus forms a narrative which sets the real, physical history and culture of Athens in a time before recorded history, whether or not there was an actual king named Theseus who ruled Athens at any point.

In Euripides’ Suppliants, Theseus declares that, “by many noble deeds I have displayed this characteristic to the Greeks: always to be a punisher of wickedness,” and Theseus in Sophocles’ OAC is a fair, unbiased character willing to offer sanctuary to the children of Oedipus, exiled from Thebes, and to hear Oedipus’s story with an open ear. He represents what Athens aspires to be as a city-state, and his portrayal in OAC may even be a reaction to the political climate of the time. The drama of the fifth century plays up (or down) the aspects of Theseus that Athenians would have known, and commentary on the state is masked behind The Athenian King’s behavior.

As to 300 BC, which is a good bit later, I can’t speak nearly as thoroughly, as that’s well after Macedonia comes in and conquers Athens. There are still mystery cults devoted to Zeus (father of Macedon), and Heracles presides over several Macedonian ritual hunts. It’s not until the third century that the Egyptian gods make their way into pagan religion as well, so Macedonian religion looks, on the face of it, much the same as traditional pagan religion.

tl;dr: A lot, pretty much, kind of, mystery cults, sermons?, probably if you want but they were kind of kooky and mostly for performing high-level rituals as I understand

If you have any followup questions on specific aspects of Greek religion, I'm glad to give it my best shot!

Sources/Further Reading:

• Sophocles, Ajax

• Sophocles, Oedipus at Colonus

• Aeschylus, Seven against Thebes

• Aristophanes, Frogs

• Euripides, Suppliants

• Homer, The Iliad

• Homer, The Odyssey

• Buxton, The Complete World of Greek Mythology, 2004

• Garland, Daily Life of the Ancient Greeks, 1998

• Henderson, The Maculate Muse, 1975 (briefly treats mystery cults in the first chapter)