For the grand canyon, I would stay in Flagstaff and day trip the area. You can stay up in Mexican hat to take in the painted desert there is/was a decent campground up there. 4 Corners really has nothing to offer IMO except the picture of being able to straddle 4 states at 1 time.
If you stay in Flagstaff and day trip you can visit Williams AF museum, ride the train to GC. You can also go to Sedona, Jerome, visit Montezuma castle, see prescott all in day trips. You can also stop in page and camp there if you want to visit lake powell.
Just over from AZ in Nev is the Flaming Gorge I think it is State park. Very nice, dont know what they have for CG's in that area. If mil retired there are nice mil campgrounds (ft tuthill in flag, and one over in nev).
North of GC there is Zion, Bryce canyon, and the state of utah has a bunch of campgrounds for the southern area. Gobblin state park, Kadachrome (named after the FILM) SP, and several others (grand escalante). We are headed that way this summer. Most have been written up in the trailer/motorhome mags this past year. If ya really wnated to you can try Jacobs lake on the north rim, but it is more isolated and a little harder to get to. Also in SO utah is cedar breaks park, not much known but pretty nice for a day visit.
There really are a lot of areas to visit, and some are not the typical tourist stops that many go to. Utah will send info if you email them, as will the rest like Idaho and Utah. If you like/want to visit some Hot Springs send me a PM and Ill give ya some place/links to look at.