Lots of good spots in Arizona. In fact, we will probably start overwhelming you with our favorites. Let me be the first, however. Mine is Usery Mountain Regional Park, just north of Mesa AZ. Also, Dead Horse Ranch in Cottonwood.
The trick is to decide what kind of "spot" you end up liking, so you should try some commercial and some state and national parks. Some of us on this forum like lots of amenities and so prefer commercial campground. Others go to the other extreme and prefer to be at least half a mile from their nearest neighbor so they prefer U.S. Forest Service campground with no amenities at all, other than a gorgeous view and lots of privacy. I am somewhere in the middle--I like electric so I can watch my satellite TV and scenery, but don't need any amenities other than cell service, paved roads, and good gravel or paved campsites because the dust drives me nuts!
Just be very careful driving down south. You are a little late in the season to be going through any high elevations because you do not want your first trip to be a slippery one. Just drive carefully. If you want help with routes, just ask.
Finally, You might want to use this site to check out campgrounds. Its pretty reliable, although you do have to read and decide if what someone likes or doesn't like is something you care about. (I could care less about dirty bathroom because I never use them, for example.) http://www.rvparkreviews.com/