I am grew up 1.5 hours north of the city and my sister lives in the city. Don't go on New Year's unless you want to just say you went.
If you go New Year's eve, get there around noon-4. You will stand in those gated areas until the ball drops. No restrooms, can't move, cops and people everywhere. If you can't see the ball they set up big TV's everywhere. Once it hits, the crowd moves and you go where they go.
NYC is great at Christmas. Things to do:
Take Staten Island ferry for a free drive by of Lady Liberty (about a mile away)
Go to Empire State building
City Bus tour (might be cold) hop on, hop off anywhere
Central Park is neat
Hit up Ground Zero
MOMA is interesting
National History Museum is huge
Brooklyn Bridge is neat to walk half way across
Rockefeller center tree and "The Rock" not as tall as Empire, but still neat.
Times Square, expensive stuff since it is all tourist
See a Broadway show, go to TKS day of and they are half off
Many great restraunts, but stuff is expensive. I like a place called Beccos. They have a special for $22 all you can eat Italian pastas. http://becco-nyc.com/
Stay at a hotel in the city, easier then taking the subway in. Expect to pay $150 for a cheap place (motel 6 or worse) and $200+ for a decent place per night.
You can fly to JFK or LaGuardia and take a train/taxi to the city for around $20-$40
Get around on the subway, only $7 for a day pass I think. Subway maps are easy to read.
Take a taxi once, just to do it, but they are pricey.
Things look close on google maps, but it takes time to travel. Walking is the best way around, not many fatties in NYC. Figure about 10-20 min per mile to travel. Farther north you go, cheaper things get.