Just going off personal experience, I've always stayed at the Super 8 in Midvale. It's way too nice to actually be a Super 8 and it's been $68 a night every time I've stayed, no matter the day of the week or time of the season. It's in the middle of every chain restaurant you could imagine so you can eat pretty cheap. Under 30 minutes to the Cottonwood ski areas (and very easy access to the bus which can be better than driving up, especially on powder days). About 40 minutes to Park City. Also about 15 minutes from downtown if there's anything there you want to do.
If you want more of a ski town experience, it's worth actually staying in Park City for a few days as pretty much everywhere else in the SLC metro is just that, suburbia.
I've heard it's one of those checklist items to stay at the Cliff Lodge at Snowbird and get "stuck" up there during an interlodge, then have the mountain to yourself for a couple hours until the roads open. Of course, that's luck of the draw (except for last year when it happened seemingly once every 3 days).
SLC is an awesome trip to make. Sure the area is very bland and basic compared to some other mountain cities, but there's nowhere that's easier to get around while having so much so close, and so cheap. Also, not sure if you're driving up to Big Sky but if you do I'd recommend also stopping in Jackson (especially with an Ikon Pass). You won't regret it.