I'd recommend visiting a ski resort. They're beautiful in the summer. Snowbird is my personal favorite, you can usually take a tram up to the mountain tops and hike down (or take the tram or ski lift down.)
Antelope Island is quite interesting, but if you're there in the summer, the little marsh bugs will be busy and there is no shade.
For food in Salt lake downtown and near public transport: The Red Iguana can't be beat for quality Mexican food, huge portions, but everyone knows it so you probably want to make a reservation. Crown Burger is a favorite of mine, they do Greekish fast food, I personally love their namesake burger with pastrami.
This is the Place is great for kids and pioneer history, the Utah Natural History museum is great also for wildlife, dinosaurs, and indigenous cultures (past and present), and Red Butte Gardens are beautiful as well.
You can take the frontrunner (train) to Thanksgiving point which has huge botanical gardens, a great dinosaur museum, and an excellent children's museum.
You could see if there are any outdoor concerts during your visit, there's a variety of jazz, rock, alternative, etc outdoor events throughout the summer.
Kind of a weird Utah specific thing is soda shops. Fiiz, Swig, Twisted Sugar...They're everywhere, they're worth a try if you are a soda drinker.
Utah is pretty serious about ice cream too. If you can get yourself to a Leatherby's, they serve up old fashioned ice cream sundaes that are delicious and absurdly big with the best hot fudge.