Standard stuff in New York is:
(1) Times Square
(2) Guggenheim Museum
(3) Statue of Liberty
(4) Empire State Building
(5) Metropolitan Museum of Art
(6) Central Park
(7) Broadway shows
(8) Radio City Music Hall and Rockefeller Square
but in general, I like doing a bit of shopping--and I like eating down in either Little Italy or Chinatown (both are right next to each other in the Soho area of New York).
For those into high-end shopping, 5th avenue is the place--and the diamond district is on 47th street just west of 5th avenue.
For one fun thing to do, consider taking a nighttime cruise on the Hudson (that goes by the Statue of Liberty), and has both dining and dancing. This is one popular New Years' Eve activity--although it can be done earlier the same week.
Enjoy
P.S. One last thing--New York generally gets less than 2 to 3 inches of snow during Christmas week--so if you want to be guaranteed a truly "White Christmas", you'd be better planning a trip to Vermont or to Colorado that week. Vermont is also where the movie "White Christmas" was supposedly filmed (although we all know it was actually done on a Los Angeles film set). There is the Von Trapp place in Vermont or you can choose from many places in Vail or Aspen in Colorado. Personally, I have spent the past 15 Christmas' in a row in New York--but then my wife's family lives there.