Arizona was made for owning an ATV. Very few restrictions are applied to ATVers in Arizona. You must have a noise muffler installed if your ATV weights in as a class II or over 800LBS.
Arizona has several National Forests that allow ATVing, but with restrictions. You must get a permit at the ranger station and you must have a registered ATV as well as being a licensed and insured driver yourself. The following National Forests allow ATV riding but under limited conditions. Apache-Sitgreaves National Forest , The Coconino National Forest, Coronado National Forest, Kaibab National Forest, Prescott National Forest and Tonto National Forest. Always check with the ranger's office before you arrive to insure the trails are not closed for any reason and the lands still allow ATVing.
If you're looking for some good ATV clubs to join in Arizona. You'll find a large number of like minded folks to ride with : Apache County ATV Roughriders located in Eagar. You can visit their website at https://www.azatvoutlawtrail.com .
Arizona ATV Riders, Inc. is another group worth hooking up with if you're in the Mesa area. You can visit their website at https://www.azatvriders.co. You know with a name like Dirty SOBs this is one fun group located in Phoenix. You can visit their website and learn more about this ATV group here : https://www.dirtysob.com.
Arizona currently doesn't require any registration of ATVs but that could change in the future. Check with your local department of motor vehicles in Arizona to insure that you're riding your ATV within the letter of the law.