I'm with the group that enjoyed road trips as a child, and don't think that even the 4-year-old will suffer too greatly from it, as long as you allow for shorter driving days and plenty of time for the kids to play. For that reason, I'd mix your campground choices between state/national parks and private CGs geared for families (KOA and a few others come to mind). Access to swimming at least every few days is a must in the summer, especially if you want to concentrate on the parks in the southwest. Yes, it cools off at night, but that camper can get plenty hot during the day if it's not being airconditioned, and it takes a while for the heat to dissipate. I would definitely recommend opting for sites with at least electric hookups. As far as kids enjoying the ride, travel games - counting animals, license plates, etc. - can fill the time nicely, and if the kids aren't prone to carsickness, some books to read or videos to watch (for SOME of the time on the road) can be good too. I might be wrong, but it seems that kids that young won't yet be so hooked on electronic stuff that the scenery and the things you see & do in the parks will go right over their heads.
With a month, you might be able to fit in not only Grand Canyon & the southern UT parks (Zion, Bryce, etc.), you could probably also fit in Yellowstone & the Tetons. I'd make sure to plan to spend at least one full day at Grand Canyon, 1 in Bryce, 1-2 in Zion (depending on the hikes), 3 in Yellowstone and 1 in the Tetons.