Southern Utah is so rich in excellent National Parks and State Parks, you could concentrate on the southern half of the state and never see it all. There are campgrounds inside the parks (book early) and commercial ones outside.
Two great resources I've used the most are Fodor's Guide to Utah, and the AAA Indian Country Guide Map, both available on Amazon.
Bryce Canyon was my favorite for the spectacular scenery, and it's at a higher elevation than some are and therefore cooler on hot summer days. At Bryce, you can walk the top of a canyon, looking down at the natural amphitheater.
Zion is the reverse, since you are in the bottom of the canyon looking up. Both are really popular and offer free shuttles so you don't need to drive, with many free ranger activities.
Around Moab in southeastern Utah, you can see Arches and Canyonlands NPs, Dead Horse Point State Park, and more. Then there is Goblin Valley SP, Capitol Reef, NP...lots of National Forest area, the list goes on and on.