The best skiing in the area is at Jackson Hole Mountain Resort, which is a 20 minute drive from Jackson. You can drive your own car or take the bus. If you are new to skiing or just want a more low-key experience, there is also the smaller (and much cheaper) Snow King ski area right in Jackson. You can rent ski gear in town (at Hoback Sports), or at the main resort (at Teton Village Sports).
Other activities in and around Jackson include cross-country skiing and snowshoeing, sleigh rides through the National Elk Refuge and snow tubing at Snow King. The organizers can also recommend good walks in or near town, and good hikes up in the national parks.