Phoenix's summary is 90% complete and summarizes the best places to stay.
The only thing I would add is McDowell Mountain Regional Park near Fountain Hills, which is near Scottsdale in the Phoenix Metropolitan area.
Campsites have water and electric and the park itself contains thousands of acres of scenic desert. We mountain bike and the trails are world class. The location is great, near Fountain Hills and Scottsdale.
The park is popular and it is likely you will spend a couple of days in overflow before getting a campsite.
Maximum stay is two weeks but you can go back to overflow and start over.