Since I am lucky enough to live in southern Arizona {25 miles north of Tucson at the foot of Mt Lemon} permit me offering some free advice.
Lots of good suggestion above but keep in mind that anywhere above 4,000' in the winter will probably remind you of... well, winter. It gets cold in Arizona and the average elevation is something north of 4,000'. Stay under 4K' and you should be fine but watch the weather carefully as it can change very quickly. Saw 6" of snow in Flagstaff the first week in May a couple years ago.
Secondly, make reservations for everywhere you go and the earlier the better. A couple of weeks is usually enough but Arizona is inundated with snowbirds from November through May As a local, we have to fight this battle for campsites all winter. I agree that the State parks offer the best of Arizona {nice amenities, weather, spectacular scenery etc., at pretty reasonable rates} so long as you can handle the crowds. Good luck!
Here is a shot of the coach parked in front of our house with Mt Lemon in the background, Catalina SP is just 6 miles down the road with pretty much the same view:
This is a shot of our backyard, again with Mt Lemon in the background:
:C
