Very few Greek plays survive to the present. I'll do my best to list them for you excluding those that exist only in small fragments. The ones in bold are more commonly produced and should probably be your next steps. The dates included should be considered approximate as some of the plays are difficult to date with precision.

Aeschylus (~525-455 BCE)

• The Persians (472 BCE)

• Seven Against Thebes (467 BCE)

• Suppliants (463 BCE)

• Agamemnon (458 BCE)

• Libation Bearers (The Choephori) (458 BCE)

• Eumenides (458 BCE)

• Prometheus Bound (???)

Sophocles (~497-406 BCE)

• Ajax (441 BCE)

• Electra (410 BCE)

• Oedipus Tyrannus (430 BCE)

• Antigone (442 BCE)

• Trachiniae (430 BCE)

• Philoctetes (409 BCE)

• Trackers

• Oedipus at Colonus (401 BCE)

Euripides (~480-406 BCE)

• Cyclops (408 BCE)

• Alcestis (438 BCE)

• Medea (431 BCE)

• Children of Heracles (430 BCE)

• Hippolytus (428 BCE)

• Andromache (426 BCE)

• Hecuba (424 BCE)

• Suppliants (422 BCE)

• Electra (415 BCE)

• Heracles (420 BCE)

• Trojan Women (415 BCE)

• Iphigenia in Tauris (413 BCE)

• Ion (412 BCE)

• Helen (412 BCE)

• Phoenician Women (410 BCE)

• Orestes (408 BCE)

• Bacchae (410 BCE)

• Iphigenia at Aulis (406 BCE)

• Rhesus (~390 BCE)

Aristophanes (~446-386 BCE)

• The Acharnians (425 BCE)

• The Knights (424 BCE)

• The Clouds (419 BCE)

• The Wasps (422 BCE)

• Peace (421 BCE)

• The Birds (414 BCE)

• Lysistrata (411 BCE)

• Thesmophoriazusae (411 BCE)

• The Frogs (405 BCE)

• Ecclesiazusae (390 BCE)

• Wealth (388 BCE)

Menander (~341-290 BCE)

• Aspis

• Dyskolos (317 BCE)

• Epitrepontes (316 BCE)

• Perikeiromene (314 BCE)

• Samia (309 BCE)

• Sikyonioi (308 BCE)

This should not be considered a complete list and some of the plays listed are, sadly, not 100% complete.

If you're comfortable with a jump to Ancient Roman plays, I highly recommend "The Brothers Menaechmi." It's a hilarious comedy of errors that tops any Greek play I've read to date.

If not, and want to dabble in the ludicrous, then try Aristohpanes' "The Birds."

The Bacchae (also spelled Bakkhai) is my favorite tragedy, and The Frogs is my favorite comedy. My degree is in Classics, concentrating in literature, so I've read a fair number of Greek plays.

Aristophanes is very funny, even to a modern reader. Some elements of humour never change.

If you get a chance, see a video of The Gospel at Colonus. It's an adaptation of Oedipus at Colonus, but it gives an idea of why Greek theatre was so emotionally involving for Greek audiences. That's something I found hard to grasp when I had only read a few plays. I saw it live at the Guthrie Theatre in Minneapolis and it was probably the most moving theatre production I have ever seen.

The Bacchae by Euripides. As E.R. Dodds tried to show it is an irrational play about rationality (or a rational play about irrationality). Either way it demonstrates that we are not all as sane as we like to think we are.

Oedipus Rex is widely considered by many historians to be the first murder mystery play in recorded history; if you've read "about" Oedipus and have not read the play, I recommend it.

If you want comedy, there is Meander's only surviving work.

http://en.wikipedia.org/wiki/Dyskolos

To get started I'd recommend particular translations rather than particular plays. Ted Hughes' version of the Oresteia is far from the most accurate, but it does the best job I know of capturing some of the power of Greek tragedy in English poetry. For Euripides I'd recommend Collier and Machemer's Medea, and for Aristophanes Douglass Parker's Lysistrata.

I read a four-play collection of Aristophanes recently: Lysistrata, The Frogs, A Parliament of Women, and Wealth. I thought The Frogs was the best of those four. It's got a great buddy dynamic between Dionysus and his sassy slave.

Antigone if you've only read Oedipus.