Since this is your first time hiking in Southern Utah, you may want to hit some of the better traveled places. I think a 10 year-old would get a thrill out of:

(1) Zion National Park
-Angel's Landing
-The Narrows, from the TOP to the BOTTOM, not the other way. It is a 13 mile hike, but not strenuous. You're in the water for about 9-10 miles of the hike. After seeing Zion canyon from the bottom, I found it amazing to see the Virgin river from the top, where it is just a creek among meadows, and then to follow it down as it carves its way through the red rock, the walls gradually getting higher and higher.
-The Subway -- an incredibly fun hike; swimming, jumping, ducking, climbing, etc. It has it all.

(2) Bryce
-For sheer beauty, this one's hard to beat. A few nice hikes in the canyon too.

Good suggestions here, Levin.

Does the campground at Calf Creek no longer exist? It's been a number of years since I last went there.

It should be noted, Quis, that Calf Creek is best visited in the spring or better, the fall. The temperatures are better those times of year and in the fall, the water in the creek is warm. The summer, IMO, is too freakin' hot.

In fact, Southern Utah in the summer heat is stifling.

It's still there.

Yes, Calf Creek is better during the spring or fall.

That's good to hear. I camped there countless times with my family growing up and need to continue with that with my family.

Anyone can do Angel's Landing, but you will need backcountry permits for the other two. You can get them online, but for the Subway you will need to enter a lottery process. If you apply during March, you can get passes for June. Apply during April for July passes, etc. You can get more info at the zion np website.

The Subway is the best hike in Utah, IMO. It's fabulous.

There are some gorgeous hikes in Arches too. There are rules about where you can camp, but Arches really works well for having a base camp and then going out on various day hikes.

Thanks for all the help
 

All the suggestions were great. I now have several good starting points.