I was in SLC this past february with a couple of friends for a senior year trip. This is what we did and it worked out perfectly.
We stayed at the Best Western Cottontree Inn in Sandy (just outside of SLC, little bit quieter, still a good number of restaurants and stuff). Every morning a bus stopped literally right in front of the hotel, and from there we could get to Brighton, Solitude, and Alta/Bird. Of those mountains, we just went to Brighton and Snowbird, both of which are incredible, and if you go out you definitely need to hit them up. We also skied Park City, but we were kinda lucky in that we happened to meet up with a friend who was there with their family, and they gave us a ride. It would probably be more tricky for you.
After skiing, if we wanted to get into SLC, it was incredibly simple. Just take the hotel shuttle to the TRAX station (the train), and take that strait into the city. really cheap and easy. A few sketchy people on the train at night but thats to be expected.
Overall, the entire trip went off without a hitch, and it was incredible. I'm sure where ever you go it would be sick, thats just how we did it. Good Luck.