Camping, per se, is another issue. Are you talking about tent-camping, or in an RV? With a tent, young kids get filthy -- and the "best" campgrounds for tents may not have facilities for cleaning them up. These types of campgrounds also are not located right next to the freeway. Most of them are off the road a few miles. When we tent-camped, we often went 20 miles off the highway to get to a decent national forest or state park campground.
If you're in a travel trailer or other type RV, you can find commercial parks a lot closer to the freeway and they'll have more facilities, too. (Some commercial parks can handle tents but quite a few specifically say "no tents".)
In either case, though, you need more time than you're giving yourself. Kids won't want to be cooped up in the car for more than 10 hours in a day, which is about 500 miles with little kids.
Really, if I were going to Madison, I'd go I-10 to I-15, north to I-70 in Utah. You could stop at Zion National Park or Bryce Canyon NP *if* you allowed yourself a little extra time (neither one are drive-throughs). Along I-70, you'll be going through the San Rafael Swell (you will enjoy) and then the Rocky mountains west of Denver, including the famous hanging highway through Glenwood Canyon. There are lots of stopping possibilities along there.
Donna