My .02 worth...
I see several people have suggested Buckskin State Park. We were NOT happy there. Water came out brown and foamy and never cleared up. Seemed expensive for a state park, especially since we didn't use the water. But, did have a hiking trail and free quail viewing.
Outside Tucson near the Desert Museum is a nice county park. (Pretty sure it was county. Wasn't Federal.)
Chiricahua was a nice park if you like easy hiking. But, sites are short. We're in a 24' rig and were parked in a (supposedly) 29' site and barely fit! Not a lot of spots there, get there early! Hiking trails from the campground and up the hill.
Karchner was good parking and a nice cavern.
BLM land around Quartzsite was pleasant. We were there "early and late" and avoided the huge crowds. Free parking. Nice desert. Hiking. Build a fire. Great place just to sit and chill out.
Bob