The Greeks wrote a great deal, and a surprising amount of what they wrote is still available to us today, 2500 years later. Their writing is traditionally divided into types:
2) the poem:
Two early Greek examples are Hesiod's Theogony and Works and Days, both from around 700 BC. There are also a number of shorter poems by Archilochus (Are-KILL-oh-cuss) and Sappho (SA-foe) from the 600s BC, among others. Sappho's poems are the only surviving literature by a Greek woman.
3) the play:
Plays are divided into tragedies and comedies. Tragedies are generally sad, while comedies are funny. The oldest tragedies that we still have were written by Aeschylus around 500 BC. We also have tragedies written by Sophocles (around 450 BC) and Euripides (around 425 BC). The oldest comedies that we still have are by Aristophanes, and were also written around 425 BC. Some later comedies were written by Menander around 350 BC. The Greeks wrote plays in verse, like poems.
4) the history:
Two major histories that we still have are those by Herodotus and Thucydides. About 450 BC, Herodotus wrote a history of the Persian Wars. About 400 BC, Thucydides wrote a history of the Peloponnesian War. After the Peloponnesian War, Xenophon wrote about his adventures as a mercenary soldier for the Persians. During the Roman takeover of Greece, Polybius wrote a History of Rome in Greek. Greeks wrote history in prose (not in verse).
5) philosophical dialogues and treatises:
The first written philosophy was written by Plato around 380 BC in the form of a kind of play, two or more people talking to each other. Later on both Plato and his student Aristotle wrote regular philosophical books, in prose without dialogues.
History of Greek Literature, by Albin Lesky (reprinted 1996).
Greek Theatre, by Stewart Ross (1999). Easy reading.
Greek and Roman Theater, by Don Nardo. For teenagers.