It's quite tough to read plays in my opinion. Hardly ever do it for fun. Would rather watch them. But I have read quite a few. Best ones: Tartuffe by Moliere, Mishima's Tree in the Tropics (don't know if that's the original title in English, have read Dutch version), Hedda Gabler by Ibsen, The Father by Strindberg, Shopping and F*cking (forgot who wrote that), Blasted by Sarah Kane and the Pillowman is a bloody good play to read as well.......
While you should certainly read complete plays, I strongly suggest the Burn-Mantle Best Plays series. You can get a great overview of a wide variety of plays over a huge period of time. If you like the overview, read the whole play.
"Inside every actor there is a Tiger, a Pig, an Ass, and a Nightingale. You never know which one is going to show up."
-John Michael Higgins in FOR YOUR CONSIDERATION