If you want to go no hook ups, Kaibab Lake CG (Nat'l Forest) is very close to Williams, off the 64 (closer than the KOA's), has lots of trees, and is pretty, and quiet. Forest camping. The upper loop is the more forested of the two. In July, the "lake" was pretty small, a very large puddle, more or less. People were still kayaking on it, but we didn't. Mid week amphitheater presentations in addition to weekend ranger talks. Some first come sites, and some of those were the best sites. Local birders put on a great talk while we were there.
We used the dump station at Circle Pines KOA (off the 40), but would not select it for a stay. The shower facilities and store were decent and quite clean, but from what we saw of the sites, gravel, uneven, sparse, and too close for our tastes. The other KOA on 64, has good reviews, but we did not visit it, so no personal obs to speak to.
If you want forest camping, try Kaibab Lake (easy access off 64) - the entrance kiosk did not stock the upper loop's map for some reason. We stayed in the lower loop a couple nights, then after checking out the upper loop, moved to a huge PT with our own piece of the forest at our site. Nice table and firering area. The recent $ investment shows. Just too bad they don't have a dump station. Some of the sites in the upper loop could accommodate 2 mh's. We're 58' total and probably had another 20-30 feet in our pull thru.
Pine Country was ok. There's a Pizza Factory in town... one of our favs. They are always in funky locations; Show Low, AZ, Orofino, ID, Bishop, CA - camping towns.