I live on Long Island, a short train ride into NYC
Some cool things to do:
Museums: ... theres a TON. MANY art museums, the museum of natural history is awesome , there is a museum for sex, the discovery channel has a museum, there (used to be, sandy took it out. Not sure if its open again), the wax museum, the tenement museum (where people lived who come over to the country in the early 1900's) if you google it I'm sure there is a museum for anything your interested.
Monuments: NYC has a ton of cool monuments. Ellis island, the statue of liberty. There is a monument for everything there. I like the Irish Hunger Memorial. The 9/11 memorial was cool too.
Food: It speaks for itself really. Many famous restaurants. You'll NEVER be hungry! Best pizza and best bagels HANDS DOWN!
Nightlife: Non-stop. I hate going out when I'm out of state. Bars and clubs stay open until 4am, some later. Whether your a have a beer and hang at the bar kind of girl or a short skirt and high heels kind of girl there is somewhere for you to hang out.
Shows: Broadway. Need I say more? You can get cheap tickets same day in times square for shows that night. I also saw the blue man group! Really cool!
Don't forget Central Park! Take a carriage ride, visit their zoo, ride the carousel, go for a walk, have a picnic... central park is lovely.
Ton's of shopping for everything from jewelry to electronics to designer bags. Everything.
Then of course there is Time Square. I don't think it's anything special, because I've been there a million times, but its really cool to see them broad cast a yankees game onto a big TV in the heart of NYC and watch everyone just hanging out in the middle of the square like it was their backyard.
China town if you want some good Chinese food!!
If you come around a holiday, like St. Patty's day or thanksgiving, or even a non-holiday, like pride month they have lots of parades.
There is TONS of things to do in NYC! My best advice is that you do A LOT of planning a head. Make sure the things your doing are close together, or you leave enough travel time. The island might only be 13 miles long BUT its very hard to get around. You'll spend and arm an a leg if you keep going north and south and north again. Plus that would be very time consuming! For example, if you want to see central park do the museum of natural history too, DON'T go downtown to see the Blue Man Group that evening. The following day go downtown and see BMG and the tenement museum and the 9/11 memorial. Catch my drift?
I'd be happy to answer any other questions you might have!