We stayed near Show Low at Fool Hollow Lake State Rec Area which I found nice but more citified than I wanted (but we also wanted showers).
Also- I see from notes from our last trip thru the area that I was perturbed by the rules of the Tonto Basin permit system on down by Roosevelt Lake. I thought my Interagency Pass would be good but we had pay an extra $15 for a daily pass to use any USFS facility.
Also- closer to Phoenix, at Lost Dutchman State Park, we learned the lesson of how a packrat can disable your vehicle in one night by chewing through wires. We started leaving our hood up after that experience but I have no idea if that works.
Things we liked in the general area were Tonto National Monument's visitor center and ruins and the pueblo at Globe.
As far as camping, I see lots of public camping possibilities on Allstays Camp and RV app.