A few more comments....

Unlike motels that are usually right off the highway, quick and easy to access once you're tired, campgrounds are generally a way off the highway. It's not unusual to drive 5, 10, 15 miles to get to a campground. Plus, there's less of them. You will likely not be able to find once quickly once you reach the point of fatigue. You must do more planning ahead.

If you are a AAA member, they have free campground guidebooks. They include county/state/national parks as well as commercial campgrounds. Or pick up a campground guidebook at a bookstore.

If I'm staying in hotels, I usually don't worry about reservations. But since there are less campgrounds, once I have an idea of where I'll probably want to stop for the night, I'll call ahead and, if a space is available, reserve it. I've done this even at 5-6pm for that night and, upon arrival, found that other campers are in overflow camping or have been turned away...which would have happened to me had I waited.

Of course, this is more true nearer tourist destination areas.

So pour over some maps and get an idea of what route you want to focus on. I-80, I-70, and I-40 are all good possibilities. Plan for six days to make the trip. And once you've made a few decisions, especially via route, come on back here and we'll be glad to offer more tips.