'Two Boots' pizza on Bleeker Street. Good slice.
A play or musical on Broadway, of course. Take a good look and you might hit tickets for a play starring a known actor - always a treat. Phantom of the Opera is pretty amazing (if not exactly an adventurous choice) and they have been playing it on the tailor-made stage in the Majestic Theatre for like 25 years.
The High-Line, an old raised train track they have turned into a garden walk up above the city. Great on a sunny day. Try the iced-coffee from a small stand under the bridge.
The MoMa too, of course. And many hotdogs from street vendors with extra sauerkraut. Clubbing in Williamsburg and Meatpacking district (prepare to spend cash here) is good. I won't get more specific with nightlife because I havent been to NYC for over a year now - try for tips from a local about what's going on. This means starting a conversation with someone cool ;)