The American Museum of Natural History. My favorite is the Hall of Ocean Life (life size model of a blue whale) and the Dinosaur (largest collection in the world) exhibits. They also have a great planetarium show but it is a bit expensive...
Check out
http://www.digitalcity.com/newyork/dining/ for a decent restaurant guide, and be sure to eat lots of good pizza!
Also, don't be afraid to ride the subway. It goes just about everywhere and is only two dollars a ride no matter how far you ride. Everyone uses the subway and it is quite safe. It runs 24 hours a day but they do get more sporadic and a bit scetchy after midnight or later depending where you are.
I love this city, have a great time!