Before my kids could ski, we often went to the mountains just to play in the snow and do a little sledding. We found that many resorts groom trails in winter so you can do a leisurely winter walk without snowshoes. This a great winter alternative to skiing for families with very little kids. You could take an all-terrain stroller. But it's more fun to take a big wooden sled, equipped with a chair and insulated stroller bag to keep your little one all snug and warm.
Below are a few winter trails I can recommend:
These are all loop trails that are manageable with kids in a sled or stroller, with restaurants along the trail, and very nice views. I'm always looking for more, so please leave comment if you know of a good one. I'll add more as I try them out.