Things get even better on 7/29, but as of now:
Open carry is the constitutional law here in AZ. Open Carry means the handgun must be in a holster on the belt, and the holster mus be wholly or partially visible. Basically, if it is obvious you are armed, it is open carry. You may carry open with "1 in the pipe" (I'd say you SHOULD ALWAYS carry that way). You may open carry anywhere except:
1. Private property posted 'no firearms' (this could be considered trespassing if ignored).
2. Any establishment that serves alcohol for on-premise consumption (bars & restaraunts).
3. Polling places, nuclear or hydroelectric plants, and schools (K-12).
4. Any federal building, including post offices.
5. State and local gov't buildings can forbid weapons, but they must provide lockers for your guns.
6. Public events (fairs, etc) that serve alcohol.
After 7/29, you don't need a concealed weapons permit to generally conceal carry, but all the above restrictions still apply to unlicensed concealed carry. There is no requirement to 'totally conceal' when carrying concealed.