I went to school in Buffalo for 4.5 years... nothing beats the cold there... but NYC in December can be just as unforgiving. Ear muffs and a nice jacket that'll cover up to your neck is highly recommended. The tall buildings help stir up the wind quite a bit. And don't forget the gloves.
Just a few things off the top of my head... for December...
- Rockefeller Plaza (the tree and skating rink)
- Central Park / Wollman Rink
- Staten Island Ferry (free) that goes right by the Statue of Liberty if you don't want to actually go to the statue itself.
- The Metropolitan Museum (The Met)
- Museum of Modern Art
- Radio City Music Hall
- Letterman
- NYSE (free tour)
- Times Square (AT NIGHT)
Another good tip (from a previous NYC thread) was to stay in a hotel outside of Manhattan. Price difference is huge. There's a nice Sheraton hotel in Flushing, Queens which is probably only a < 30-minute subway ride into Manhattan anyway. Depending on how much you want to spend on the trip, this would be a good way to save cash. Besides, you'll be taking the subway around the city somewhat to get to all these sites... so why not. They have full-day subway passes for $7/$8 I believe.