Views from the top just had a thread about winter hikes for beginners, mostly in the Whites:
The AMC lodge at Pinkham Notch, NH, is a starting point for a number of hikes. Don't go all the way up Mt. Washington unless you're well prepared, but you could ask the staff there. Hiking into Tuckerman ravine is interesting (follow avalanche warnings) and Boott Spur is interesting.