This highly depends on what you're looking to read. I'll refrain from posting specific editions or translations, since they of course highly differ from one language to another.

For relatively easy to digest philosophical writings I recommend:

Meditations, by Marcus Aurelius (Yes, the emperor!). Literally "Writings to oneself", these are Stoic musings on various topics. Thought-provoking and requiring not too much expertise to start with.

Enchiridion, by Epictetus. A collection of various thoughts on how to lead a morally correct life, also Stoic. Very appealing due to its highly practical, and oftentimes still relevant nature.

If you're looking more into Plato, I'd recommend the Symposium. A highly influential text on the nature of Love. This one is a little harder to fully comprehend on a first read, since the conclusion wanders of into a highly metaphysical concept of the Ideas of Love and Beauty, and the nature of the true philosopher.

For tragedy, I would recommend Euripides' Bacchae. It's a tragedy about the return of Dionysus to his place of birth, Thebes. The king rejects him, and Dionysus turns on the town in a bloody wrath, ultimately destroying the king and Theban society as a whole.

Also noteworthy is Sophocles Oedipus King, but Sophocles tends to not go over very well with modern audiences, since he very often incorporates very explicit moral instruction in his dialogues, which modern readers aren't very fond of.

For rhetoric texts, there is a marvelous little jewel of a text: Gorgias' Encomium Helenae. This is a short speech where Gorgias attempts to prove that Helen of Troy was a morally virtuous woman, in spite of what was commonly thought. This is an especially interesting text if you read the original Greek, since every sentence abounds with wordplay and stilistic bravoure.

For historiographic texts, I highly recommend Herodotus' Histories. The project of this work is to examine how the feud between Greeks and Eastern barbarians (mainly Persians) came to be, and also includes many, many historic (or rather, pseudo-historic) asides, explaining local customs or telling local tales. Very entertaining, though best taken with a grain of salt.

Also Plutarch's Parallel Lives are extremely good. These are relatively short biographies of virtuous Greeks and Romans, always in pairs. One good example is a parallel biography of Caesar and Alexander. The Lives should be taken with a grain of salt as well: Plutarch's main goal was not historical accuracy, but rather moral education.

For epics, of course, Homer's Iliad and Odyssee. The Iliad focuses on the consequences of Achilles refusing to fight in the Trojan war because of a slight inflicted upon him by Agamemnon.

The Odyssee focuses on Odysseus returning home from Troy, and his son looking for him. The Odyssee tends to appeal more to modern audiences than the Iliad; it has more aspects of an "adventure story" rather than a series of political debates and battle scenes on the shores of Troy.

Also an incredible epic story is the Argonautica by Apollonius of Rhodes. This is the story of Jason and the Argonauts reclaiming the Golden Fleece from the Eastern king of Cholcis. A main theme is the love of Medea, princess of Cholcis, and Jason's betrayal of her.

If you would happen to have the chance of reading the original Greek, I cannot recommend the archaic lyric poets highly enough. All of them have only been preserved in fragments, but those remaining fragments are often extremely beautiful. A special recommendation is Alcman.

That should keep you going for a while. If you have any questions on a specific genre or work you'd like to know more about, you can always shoot me a PM.

And lastly, I'm sorry if my English isn't 100% perfect; it's not my native language. But I couldn't pass up this opportunity to finally contribute in my field of research!