Zion National Park
We camped at this park for three nights and really liked it. We stayed on the Watchmen loop although both the South loop and the Watchmen loop were nice. There are no showers at this park, but they do have clean drinking water and bathrooms.
My favorite part of this park was hiking the Narrows, although I will say that this is NOT a kid friendly hike unless your kids are able to hike on their own and are good swimmers. You hike up a narrow canyon through a shallow running river. The scenery along this hike is breathtaking. It is a total of 16 miles long and takes about 13 hours to hike in it's entirety. There were plenty of backpackers that were going the entire way and camping as they went, but we only hiked about 31/2 miles in and then turned around and went back out. I would definitely love to go back and hike the entire length one day.
Bryce Canyon National Park
Even though we didn't spend the night at this park, I felt it deserved a shout-out. This is by far, one of the neatest parks I've ever visited. When walking through it, you literally feel like you've entered The Flintstones. It's not a big park, but it's definitely memorable. You can see most of it in one day if you're not planning any major hikes.
While we didn't experience either of these parks with kids, they are both kid friendly and offer shorter hikes for the younger crowd in addition to junior ranger programs for kids aged 6-12. These parks are located in Southern Utah and you can learn more about them at their websites: ZION and BRYCE.