New York City is amazing, there's nothing you can't find here. You can walk and take transit easily to anywhere in Manhattan. The Met is amazing, and you can choose how much you want to pay for entry. Central Park is gorgeous when you want to get your nature fix, or go to the long stretch of park running along the Hudson River on the Upper West Side. It's very easy to grab a cab when you're in a hurry, so don't be afraid to hail one.
I would definitely recommend the Staten Island Ferry, which is free and
Insider Tip:Save money staying at a hostel that's central to Manhattan. You'll save a lot of time and headaches getting into the city, and enjoy easy walking/bussing/or metroing back each evening. I really liked the International Student Center, for those under 35. Buses are extremely easy to figure out, as they run pretty much north/south and east/west, and run very frequently. Definitely buy a Metrocard and load it with at least $10 for easy trips home when your feet are tired. You can add value at any Metro station