From the base of little cottonwood to park city is a 45 minute drive. You absolutely must go to Alta and Snowbird though, unless you are a park rat and prefer park city. They get a lot more snow at alta and snowbird and the snow is lighter cause they are at a higher elevation by a couple thousand feet i believe. If you go to Utah, it is the back country and steeps and cliffs and that scene that you MUST experience...ski park a little, but you can go elsewhere for that.
I recommend right under the supreme lift, alf's high rustler, and definately devil's castle for some great in-bounds skiing.
Snowbird's mineral basin is also amazing when it isnt skied off, so if they get snow its great. Also hit up gad chutes, upper cirque, and i think its called peruvian gultch or something for some good stuff at snowbird.
Another good mountain is in big cottonwood and is a lot less known than all the other mountains...solitude. good skiing, last time i was there, weak park.
We'll have you dead pretty soon.