Snow shoeing would be a must for me. There should be enough around Jackson, and Bridger-Teton NF around US26 to the northeast will have more trails.
Keep in mind a lot of the places up that way can get bad air pollution during the winter due to the cold (high pressure air and temperature inversions), high vehicle traffic (especially snowmobiles), and bowl-like topography. I don't know how bad Jackson Hole gets, but West Yellowstone, MT is notorious for this.
Jackson has plenty of good bars and restaurants, plus a few museums if you need something indoors. The elk at the refuge should be easy to spot since they'll likely be out of the trees and in the grasslands. Might see some bison out there, too.