If you weren't planning on renting a car, you may want to stay slopside. I don't know of any hotels that are going to offer shuttles to the resorts, and there are other shuttles, but I believe that you're gonna end up paying as much for them as you would to rent a car. (I'm not positive on this one but pretty sure.)
If you stay in SLC, you will be able to ride public transit to any of the cottonwood resorts (Brighton, Solitude, Snowbird, Alta) but I would not call that an "efficient" shuttle to the resorts.
Check out skiutah.com, or go-utah.com, they can give you some more infomation about the transportation issue, but I think if you are going to be staying in SLC, and going to Park City / The Canyons, renting a car would be the best way to go.
I live near SLC, so I don't know the details of transportation from Ogden to Snowbasin / Powder mountian, but I would guess it is the same.
All the hotel chains have sites in and around SLC, you may want to check with them to see if they offer shuttle service to any of the resorts.
I believe all of the hotels in and around Park City will offer some kind of shuttle service to the resorts - again check with them and see. There are a few that are just outside Park City that may be more affordable.
OK, I ranted forever, hope this helps...
Oh, and if you're into powder, check out cat skiing....