The painted desert is a kind of big area with few facilities. A lot of it is on either the Navajo Reservation or the Hopi Reservation.
That said, there are highways leading through, and they're visually well worth the mileage. Personally, I'd recommend something like a little round daytrip from (near) Winslow, up to Second Mesa, then around through First Mesa and Keam's Canyon, back down south to Holbrook, then interstate back to Winslow. Or vise versa. End your day with a nice meal at the train station restaurant in Winslow! You'll never forget it.
In Winslow, we generally stay at the nearby state park, Homolovi Ruins. But there are some RV parks around, too.
๐Lynn