You've received plenty of great ideas, but I thought I would add just 2 more along with the link so you could reserve your spots. First is Payson Lakes (Alpine Loop, near Maple Bench that someone else already suggested) has always been a favorite of ours. The campsites are nice and the lake has a paved path all the way around, which is great for going for walks with little ones, or riding bikes, pushing strollers etc. There is fishing if you're interested and a grassy "beach" with picnic tables and BBQ grills. Second is Spruces in Big Cottonwood Canyon. Very well maintained campground, just 9 miles up the canyon. Both of these campgrounds can be reserved through www.recreation.gov as well as several of the ones that others have suggested to you. Hope this helps! Also, our all time family favorite is Moon Lake (near Duchesne, UT), but this is a 2-3 hour drive. Definitely worth checking out if you're ever up for a longer trip. We go there every summer!