The DEC has a list on their website of "Great Winter Hikes". It has a few local sites listed including: Five Rivers in New Scotland, and Partridge Run in Berne. Other recommendations, Buck Mountain and Camp Santanoni, are also not too far away.

Check out the list here.

From my own experience:
Five Rivers has nice flat terrain for snowshoeing (you can rent shoes there) or XC skiing.
And Buck Mountain has a steady ascent with a very steep/flat rock portion at the end toward the top. I've never done it in the winter but I hear it is challenging, but fun.

Enjoy the winter weather out there!