This is in addition to the excellent recommendations for museums/galleries etc above :)
The last time we were in NYC we were given a great tip by a New Yorker, to visit the observation level at 'the top of the Rock' (The Rockerfeller Centre) instead of going up the Empire State Building. You get the classic New York view to one side, which includes the Empire State Building and then stunning views of Central Park from the other side. Minus the extensive queues that The Empire State Building attracts.
Visit China Town, Little Italy, the Statue of Liberty (not Ellis Island if you're pressed for time - lots of school trips!) and take a pic at the Flat Iron (first skyscraper). Also if you're a Sex & the City fan (don't judge me), get a red velvet cupcake or the banana pudding from Magnolia Bakery and have steak frites in Pastis (if it's reopened). Also, The Mandarin Oriental hotel has a terrific view over Central Park for cocktails.
We also loved an evening at the Comedy Strip Live (this was where Seinfeld, Chris Rock etc got their breaks).
We LOVE New York! I always say that if I could live in another City for a couple of years, New York would be it.
Enjoy! x