I live very close to the city, so I get word of new stuff and reviews pretty quickly. I'm only going to speak of musicals here, because I'm not very familiar with the dramas. I've heard The History Boys and Faith Healer are supposed to be fantastic. And of course Three Days of Rain- but tickets to that might be impossible to find.
I would highly reccomend Drowsy Chaperone, The Light in the Piazza, and Sweeney Todd. If you MUST see a jukebox, go with Jersey Boys- but I for one don't like them. Putnam County Spelling Bee is also fantastic.
The only show I wouldn't spend my money on-
Tarzan has received reviews that all but set fire to the theatre. The critics, my friends, and pretty much everyone I've spoken to say the set is fantastic, everything else, on the other hand, stinks on ice.