Hmm, i went to NYC once and i'll tell you some things that girls that live in NY told me and some i experienced:
1- Don't spend your money and time climbing up onto the Empire State if it's winter - it's too cold and windy (as i read) or the Statue of Liberty - the ferry that goes to Staten Island is free/very cheap (don't remember right now because we didn't go there because it was too hot on that day and where my mom's friend used to work had a conference room with view to it) and it passes right beside the Statue - the trip to go there costs around 20/25 buck and everyone tells that it's not worthy
2 - Battery Park - it's very beautiful and neaby there's the Financial District - if you like to see buildings , there's a lot of beautiful ones . Plus going down (somewhere i don't quite remember) you get to the South Sea Port (there's a small mall there) and you can see the Brooklyn Bridge
3 - Central Park! Check out the area where is the homage to John Lennon.
4 - MoMA if you like art - it isn't ginourmous and overwhelming like the Metropolitan, and there's a lot of important and cool art exposed there (and try to fight the urge of touching the Starry Night painting - it has such a cool texture)
4- A place that i went and found that is cool is the Chelsea Piers - although we didn't go there at day, i've seen in many place that there is a sports area and etc.
5- Take a small trip to Chinatown, Little Italy (that nowadays is just MulberrySt.) and Soho.
Too many things to do, such a short time!