Five National Parks here in Utah - so feel free to swing by this state. Capitol Reef National Park is probably one of the least explored, especially the northern end. In that end, all of the roads are unpaved, and if I remember correctly, they recommend a high-clearance vehicle.
And there's a nice RV park in Torrey, UT...
Our site at the RV park (back in 2008)
View from the RV place shortly after sunrise
In fact, if I remember correctly, the majority of the roads in Capitol Reef are not paved.
Zion National Park is pretty much all paved - and during the higher visitation months, you've gotta use their shuttle system inside the main part of the canyon.
Bryce Canyon is interesting. We'll be there for a week or so at the end of June / first of July.
Moab area? Yeah, you've gotta do that one! Arches and Canyonlands National Parks, as well as other interesting areas close by.