Drive up to Park City for the day, hike around the paths. If you're insistent on staying in the city, visiting Temple Square and the complex is neat and beautiful even if you're not LDS. Millcreek Canyon is one of my favorite parks. Although, I don't recommend buying anything, being Mustachian and all, the City Creek Mall is neat to visit and walk around.
My kids loved visiting the zoo, the library, and the children's museum.
I also second visiting the Red Iguana (with a kid, I actually find the Red Iguana 2 a little easier as it's more spread out) but get there early. Great Mexican food and Moles.