Take the trails out of Alton that go right up the back side of Mt Major. Nice riding in there and not busy either. The trail head is right near the boat ramp in Alton and works it's way between the houses. Once you hit the high country there are some great places to take in some views!
Corridor 22 is a nice ride heading east into New Durham and eventually out towards Brookfield. Snow conditions are everything though.
Everyone would say take her up the Castle trail but that when the conditions are reasonably good are pretty busy, not best for a beginner! It's a great ride but you'll want to get used to riding and handling a sled before going out in an area that's crowded. Some riders can be impatient with beginners! Once you have the hang of it defiantly do that ride provides nice views!
https://maps.topographics.org/hosted/nhsa/
The above link you can zoom in and see the primary trails. The trail I'm speaking of is NOT on this map.
If you want to take a little ride, the Baker River Valley system is awesome, they are about a 40 minute ride from Alton. Was there last weekend and it was pretty epic! Funny that area was BURIED in snow and everyone is driving all the way up to Twin and Pittsburg!