There's also the Rockettes Christmas show at Radio City Music Hall. In early December I love to just window shop on Fifth Ave w/ all the Christmas displays.
If you're into museums, I highly recommend the American Museum of Natural History, esp the Rose Center (Hayden Planeterium). I'd also recommend purchasing the membership for a nominal fee, it allows you to bypass the traditionally long lines (on weekends).
http://www.amnh.org/rose/?src=h_p
Other points of interest are the Intrepid (aircraft carrier), Empire State Building (observatory is now re-opened), Ellis Island / Statue of Liberty, and Greenwich Village.
Where are you going to be staying? Times Square is the heart of Manhattan, but pricey too. Obviously there's Broadway shows there, the TKTS booth in the heart of Times Square has 1/2 price tickets the day of performance, you usually have to start lining (queuing) up about 10am or so.
I'd suggest purchasing a Fodor's or other type of guide as it's such a big place, it can help. Worth the investment. Zagat's will also give you a good restaurant perspective.