You have a good idea for your stay if you need to keep a tight budget. The bus links to both Cottonwood Canyons from the south valley are great. Motels in the South Town Mall (?) area in Sandy are well located. I would suggest going to the UTA website for the exact routes and times. You also have the option of skiing Alta, Solitude, and Brighten just as easily as Snowbird. Also, there are a number of ski shops in the area. Check with them and the motel for discount lift tickets to save a few more bucks.
I have done this kind of trip twice in the last four years and it works out well. I stay at the sister-in-law motel across the Interstate in So. Jordan, and get a ride to the bus stop ea morning. It's not exactly that mountain ambience, but if I had to pay for that I couldn't afford to go. I was just checking the airlines today for a mid Feb trip. Good luck.