First thing is you need to establish "residency" in AZ. You do that by having any official document mailed to you at an AZ address you are now calling home or you present any acceptable evidence of residency issued from any entity that established your residency such as a water, gas, electric bill or a residential lease for your AZ address. Basically you need some proof you live in AZ or otherwise meet residency through employment or government service.
With that, you go to any MYD office and apply for a regular drivers license.
If you pass the tests (written and eye), they take your old license from out of state, void it and issue you a temp AZ license. With that temp, you now apply for registration/titles for the vehicles.
In AZ, there are also authorized third party offices that can pretty much do all the same things just you pay them a service fee. Their offices are never crowded and much easier to deal with. With the exception of Pima and Maricopa county, you can register your vehicles in any county regardless if you reside in that county or not. Pima and Maricopa have emissions testing so you you must first take it in for testing before registering if the address in in those counties.
Now, last year I know they started cracking down on mail service and private mail box services for address. You must have a legitimate street address even if using a box number for mailing purposes. I can;t find much details just that people have been turned away due to a "illegitimate residential address". You may want to call to get the details. Some RV mail services will contract with an RV park for your "address" to be at the RV park.