Since the OP was talking about getting to Epic/Vail resorts in UT, I think he means Canyons and Park City. Using the link provided by litterbug, it does look possible to get from SLC airport to/from Park City Canyons Transit Hub and Park City Transit Center for $7 using Public Transportation - see http://www.parkcity.org/index.aspx?page=1032 for the PC-SLC Bus Connection. The caveat is that you cannot pack may than you can carry on and off light rail or bus and service from downtown SLC to Park City ($4.50 - exact change) is very limited, as it's only available early and late in the day (peak commute times) since it looks like it is for people to get to and from work.
When I flew into SLC last Feb, I took TRAX ($2.50 can use credit card) to downtown SLC, walked around as much as possible while carrying skis and dragging roller duffle with my boot bag on top of it. After making sure I knew where to catch first late afternoon commuter bus to where I was staying in Cottonwoods Heights, I went into bar, leaned skis and bags against wall and had a beer. One guy getting on bus complained about how much space I and my bags were taking as I think I was in his usual spot. Bus driver was very accomodating and would have dropped me across street from friend's house, but I was OK with very short walk across parking lot.
You should be able to get around Park City without a car. But having one is recommended if you want to go to UT ski areas in the Cottonwood Canyons, where they get more snow and usually have much better snow conditions in the spring.