One of the best things to enjoy while you're in London is the theatre, but most of all musicals. I have been to some of them (you can choose from a literally endless list) and I have loved them! Actors are really talented and with amazing voices and sets are incredible. Check them out, it's really worth it. And don't be afraid if you are a student or you can't spend a lot, you can easily go to each play with 20/25£. Even if not so close to the stage, you'll enjoy every detail anyway, I assure!
Here some web sites where I use to buy my tickets:
Lastminute (even if you are booking in advance)
I guess this post works for New York too!
(from Rivista Idra)
(from Mariano Diotto)