Since Arizona is home to the world's largest gathering of gem and mineral dealers (Tucson hosts 40+ shows January to mid-February) and a huge outdoor "rock swap" (Quartzsite, January), the state's best rockhounding is probably done with a wallet instead of a shovel.
But if you're interested in the "do it yourself" approach, you can collect "reasonable amounts" of minerals, rocks and gems on most Bureau of Land Management property -- for personal use only, not for sale or barter.
Start by checking land ownership at the local BLM office. If you can point to a location on a BLM topographical map, they can determine if the site is on public lands and open for rockhounding.
More info: Bureau of Land Management, 928-317-3200.