Day #1: Your favourite play
This is not very difficult: it's Hamlet. To repost most of what I said about it a few years ago, this is pretty much the pinnacle of Shakespeare's literary powers, and has been rightly regarded as such for centuries. A lot of this is because of the fascination of the central character, advised of his father's murder by his father's ghost, and then taking a troubled but compelling path to vengeance, which ends up not only with his own death but also those of his father's murderer, his mother, Polonius and both his children, and Rosencrantz and Guildenstern.
Also, of course, the language is amazing. This play surely has more famous quotes per page than any other, most of them short phrases that neatly bracket some concept - "a consummation devoutly to be wished", or "the slings and arrows of outrageous fortune". It's occasionally rather startling to hear the original context of some commonplace line, though it doesn't really jar the play.
Apart from the main plot, I find two interesting themes in the play. One, not surprisingly, is death. Everyone is talking about it, from the king to the gravedigger. Depending on how you count Julius Caesar, this is the first non-historical play with a ghost. We end up with the stage littered with corpses, and I think there are more on-stage killings than in Titus Andronicus - and unlike Titus Andronicus it isn't over the top. (It's also difficult to deny that there must have been some connection in the author's mind between the title character and his own son Hamnet, who had died a few years earlier aged eleven.)
The other theme I picked up was the theatre. It's not just the play-within-a-play (though that is more interesting here than the comedy of A Midsummer Night's Dream, let alone the peculiar unfinished framing narrative of The Taming of the Shrew); it's the conversation of the players with Hamlet before the show, and the final discussion between Fortinbras and Horatio about telling the story and displaying the bodies. Shakespeare isn't overdoing it, but he does seem to want to make us think about what the theatre is and what is happening when we are watching (or in this case listening).
I first saw it as a (somewhat cut) school production in 1979 when I was 11 or 12. I think I may have seen a student production in Cambridge as well. I enjoyed the David Tennant version (and I want to do a fanvid some time cross-cutting between him and Derek Jacobi in the 1980 version, where Lalla Ward was Ophelia and Patrick Stewart was Claudius).
But I want to particularly mention a great stage production I saw in Brussels 18 months ago, "Hamlet Unplugged", which had just four actors, each playing Hamlet in turn, and speaking only Hamlet's lines, everyone else miming the other parts. It was brilliant.
The 30 days:
Day #1: Your favourite play
Day #2: Your favourite character
Day #3: Your favourite hero
Day #4: Your favourite heroine
Day #5: Your favourite villain
Day #6: Your favourite villainess
Day #7: Your favourite clown
Day #8: Your favourite comedy
Day #9: Your favourite tragedy
Day #10: Your favourite history
Day #11: Your least favourite play
Day #12: Your favourite scene
Day #13: Your favourite romantic scene
Day #14: Your favourite fight scene
Day #15: The first play you read
Day #16: Your first play you saw
Day #17: Your favourite speech
Day #18: Your favourite dialogue
Day #19: Your favourite movie version of a play
Day #20: Your favourite movie adaptation of a play
Day #21: An overrated play
Day #22: An underrated play
Day #23: A role you've never played but would love to play
Day #24: An actor or actress you would love to see in a particular role
Day #25: Sooner or later, everyone has to choose: Hal or Falstaff?
Day #26: Your favourite couple
Day #27: Your favourite couplet
Day #28: Your favourite joke
Day #29: Your favourite sonnet
Day #30: Your favourite single line