I just barely moved away from Utah, and I'm already missing it. I don't think the skiing would be that good at this time of year, but definitely a must-do in Utah. The slopes are amazing.
During the summer, there's plenty to do still. You may want to hit the "Alpine Slide", or go to Hogle Zoo. If you go to Hogle Zoo, right up emigration canyon is an awesome diner, "Ruth's Diner". It's been awarded best breakfast in Utah by our City Weekly magazine for like 7 years running. Great restaurant, even for stuff other than breakfast.
Also, you may want to take a walk around "Temple Square". You can see the Mormon's headquarters.
The LDS religion is an integral part of Utah history, and is interesting. Plus, the missionaries are all out in the field, so you can actually have a peaceful walk through the area.
Stare at the mountains as much as possible, they're a beautiful sight. Sunsets & Dawn are accentuated by the mountains in an amazing way.