You don't mention whether you are looking at a backpack, or basecamp doing day trips. Although a bit further away than the Uintas- I would go for Capitol Reef, especially for day trips. Will be somewhat busy around Fruita, but in my experience at that time of year- uncrowded once a little ways out on the trails.
Oops, that information might have been important. I am looking to do a backpacking trip. I think I might look more into the Uintas since it is the closest to where we are going to be. I did just a quick google search and found some recommendations for Glady's lake or Red Castle Lake.