Central Park is worth a visit at any time of year. Times Square at night is a must-see. Both are free.
If it's so cold/rainy that you need to stay indoors for a while, The Met (Metropolitan Museum of Art) would be my first choice. It's on the Eastern edge of Central Park and is massive. Paintings you've seen in textbooks, an entire Egyptian temple shipped over and reconstructed, and beautiful architecture in its own right. Student admission is $10 (bring your student ID).
See http://www.ny.com/transportation/ellis.html for info about cheap ferry rides out to the Statue of Liberty. Nothing can beat taking your very own pix of her.
Spend 5 minutes looking at this map (maybe print it and bring it with you): http://www.aaccessmaps.com/show/map/manhattan.
- the various neighbourhoods in relation to one another
- how the streets start in the low numbers at the bottom and get higher (which is why people talk about lower and upper Manhattan)
- how there's a West side & an East side. The dividing line - depending on where you are - is 8th Ave/Broadway/5th Ave
(This second map is somewhat easier to read. I had to tell my printer to reduce it down to 60% to get it on one page.)You can get around by foot, subway, and bus. Because there are interesting buildings/restaurants/shops around every corner, New Yorkers walk a lot. They're among the most fit people in America.
To use the transit system, go down into a subway station and buy a MetroCard (transit card). Fare is $2 a ride. If you load $20 on the card, they give you $23. More than one person can use the same card. You swipe the card when boarding a bus as well as in the subway. Bus and subway maps are here: http://www.mta.info/mta/maps.htm
If a few of you split the bill, taking a ride in a yellow cab - even just for 10 blocks - is a quintessential, inexpensive New York experience. My first time there, I was astonished by how many yellow cabs there are.
Also, before you go, check out:
this list of museums & cultural institutions
this list of famous buildings, sites & monuments
Click anything on these lists for more info.
You're gonna have a blast!