Chiricahuas east of Tucson are worth a visit. There is boondocking along Pinery Creek just south of the monument. If you are lucky in the Chirachahuas, you might see coatis.
There is a really sweet camping site at the Verde River Sheep Bridge in the Tonto NF. Long drive in (36 miles) and the last three miles are pretty rough, which is good as it limits use. Bloody Basin Road, just south of Cordes Junction on I17. Takes you through Aqua Fria NM.
if you are into funky old mining/now artists towns, both Jerome and Bisbee are worth it. Bisbee is south of Tucson and on the border. Copper mining. Jerome is south of Flagstaff. Silver mining. I prefer the back road from Williams to Jerome which takes you up the old rail bed. Most people drive up the paved road from Clarksdale.
if you are into astronomy, there is Lowell Observatory in Flagstaff. Lots of the boondocking sites closest to Flagstaff have been closed by the Forest Service in the last few years, but there are plenty of possibilities a little further out. For museums, the Museum of Northern Arizona has a fine collection of native artifacts.
On 180 between Valle and Flagstaff, there is Red Mountain. Geologically interesting as it is a volcanic cinder cone with one side sheared off. Short hike in.
East of Flagstaff is Wupatki/Sunset Crater. Wupatki has a host of native ruins, while Sunset Crater has a host of volcanic features to explore. Also Walnut Canyon for Native ruins.
The rim drive along the Mogollon Rim east of Clints Well offers nice views. You can camp along the rim.
We're looking at the other suggestions. 
