Salt Lake is the first one that comes to mind for me too. But I would skip Park City, just stick to the Cottonwood Canyons areas. Park City still requires some driving, Cottonwood Canyons are right there...
I forget your ability level. Are you intermediates? If yes, go to Solitude, Alta and/or Brighton. Those three have the best selection for intermediates, meaning extensive and varied too. Alta, of course, just oozes the history of skiing. If anyone in your family only snowboards, you'll have to skip Alta because they don't allow snowboarders. And if anyone in your family does snowboard, you really should consider a day at Brighton because it has a very cool "snowboard" scene. Brighton also has wonderful intermediate trails snaking through the trees, more like an Eastern resort than a Western one. We think of it as "Jay Peak on steroids."
Solitude is aptly named, so if you really want to avoid crowds, go there. All three resorts have excellent advanced and expert terrain as well, if you have anyone of that ability level in your family.
I really don't recommend Snowbird for beginners/intermediates unless you are truly adventurous. The mountain can be very intimidating. And the trail choices at those ability levels are limited.
If you stay at certain lodgings, you can get a multi-day ticket that's good at all the Cottonwood resorts, so you can explore as many as you want or just stick to one (it's nice to have that option).
Thatsagirl
