DD lived there for a year, so if you have any specific questions, do ask away.
First off, buy a pop out map, they're brilliant. Then start with a hop on hop off bus trip to get yourself acquainted with what's where. After that, its about what you want to do really. Definitely recommend the ferry out to Liberty Island/Ellis Island. Also the 9/11 museum which you could do on the same day as they are near each other.
Central Park is great, but huge - you could spend a day walking there, the zoo is depressing so I wouldn't bother with that. Unless you want to visit the M & M store with children, Times Square is a waste of time - loads of tourists milling around some billboards, nothing else to see!
We did the Empire State (pre-booked) last time, which gave us fantastic views, but I know others have said The Top Of The Rock is better. If you want to go to a show, you could get discounted tickets from one of the TKTS booths, although we were lucky enough to get some huge discounts with a coupon from the immigration officer who checked us at the airport!
Food is expensive, and most of the tourist traps aren't cheap and not great - eg Ellen's Stardust Diner, but if you head away from the most touristy areas, it isnt too bad. Lots of cheaper restaurants in Lower Manhattan. We tended to eat a big brunch and then again in the evening, didnt need lunch.
We used the subway to get about, it was fine (not very clean though), but we did come across some strange characters on the trains. Not in an unsafe way, just a bit unusual, all part of the fun of holidays though.
Am sure you'll have a fantastic time, am very jealous.