In Glacier National Park in winter, Going-To-The-Sun Road is typically plowed from the west entrance to the Lake McDonald Lodge parking lot. Easy strolls, snowshoeing or skiing can take place up Going-To-The-Sun Road from here or along McDonald Creek. More ambitious snowshoe or ski hikes can take place up the Sperry Chalet and Mount Brown Lookout trails.
Apgar village is plowed out in the winter (one shop stays open all winter) and from Apgar winter trails lead to the Fish Creek and along McDonald Creek (the Oxbow Trail). On weekends, park rangers lead two hour snowshoeing hikes from the Apgar Visitor's Center.
The Camas Creek and Polebridge entrances to Glacier National Park are open to foot traffic in the winter but deep snow usually makes snowshoes or skis a must. No worries, you can borrow the snowshoes or skis from Moss Mountain Inn.
Near Moss Mountain Inn are many Flathead National Forest hiking possibilities also, depending on the snow cover, the amount of time available, and your level of fitness.