In Brooklyn:
I thought the pizza at Di Fara was delicious and Lucali a close second.
When you're in Manhattan:
Definitely head to the top of the Rockefeller building. On a clear day, the view is incredible and you can take photos with the Empire State in it.
The size of the Rockefeller ice rink is much smaller than I expected, but it's one of those iconic places you always see in movies/tv shows. If you have a chance, rent some skates and go for a spin. It's a neat experience.
If you have the time, do the tour of MSG. It's pretty inexpensive -- maybe $10/$15 from what I recall. The building has so much history and the tour guides have some great stories to tell. Highly recommended.
Most importantly, enjoy yourself
It's an unforgettable city.