Another vote for State & National parks, and Corps of Engineers (always on the water.) You will see more of the "real America". Not that private campgrounds are bad. We have stayed in several KOAs and a few casino resorts which were very nice and had a lot of stuff for the kids to do. But my preference is always for the more wooded, rustic campground where I can see trees and have a little campfire if I want to.
We took our kids on a cross-country tour a few years ago, when they were 12 & 14. Two months in a 19ft trailer--we got to know each other VERY well, LOL. The only reservations we made ahead was in Yellowstone, because we were going to be there in July and wanted to be in a specific campground near family members. Otherwise, around 2-3pm we'd figure out where we expected to be at 5pm, looked in Woodalls and RVParkreviews and made a few calls. We like to get off the road while it's still light out, so the kids can get out and run around, go to the pool, etc, and I can cook dinner without the rush, and even get a load of laundry done. Only ONCE in 60 days did we come across a sketchy campground and that was totally our fault. We didn't follow our rule about getting off the road by 6pm. Tried to "make a little more time" and ended up driving in the dark. At that point we were running out of options, so we pulled into the only campground available.(we don't boondock with the children). It was a total dive--we didn't even have 30amp electricity! Thankfully it was cool enough to open the windows. We got up early and pulled out of there by daylight.