I can recommend the public transportation (BUS) to the resorts. It's only a couple of dollars each way. It's NOT like riding the bus elswhere, even in SLC(ie Rose Park). Everyone is going skiing. There is a "DIRECT" bus that goes to Alta/Snowbird from right infront of the Red Lion(a couple in the morning). If you don't catch the "direct" bus, you can do the Trax/Bus combo. I don't recommend it, but it works on the way back if you're in a bind. The stop is actually in front of a gas station to the side of the Red Lion. There is transpo to the other resorts except for the Park City area.
I really see little need for a car in downtown SLC. The have the trax system that is free within the downtown area. It's really hassle free. Done this scenario many times and friends have too.
Best of luck and happy skiing. I wish I could be there this winter.