Welcome to AZ
Folks often post questions about Shooting in AZ on forums, I often reply, I finally saved my replies so I won't have to retype everything each time. This probably gives more info than you asked for but its easy to copy/paste. latest update in Aug 2021.
Websites:
http://www.arizonashooting.org/ web group
http://www.azshootingzone.com/forums/ web group
https://arizonagunowners.com/ web group
https://www.ar15.com/forums/hometown/Arizona/13/ another web group
http://www.phoenixrodandgun.org/ A club with a range in south phoenix
http://www.rsscaz.com/ A club with a range in east Mesa
https://www.azgfd.com/shooting/basf/ A range with clubs in north phoenix.
all 3 have public ranges and competition ranges and active competition groups
http://arizona-rifleshooting.com/ my website about competition
3 ranges in the Tucson area:
http://www.tucsonrifleclub.org/
https://pimapistolclub.com/
https://tucsontrapandskeet.com/
orgs:
http://www.asrpa.com/ our state association they do: competition, training, education, run a range west of phoenx and civil rights advocacy.
http://www.azcdl.org/ is 100% civil rights
NRA/CMP/PRS style competition groups:
https://phoenixrodandgun.org/ (prgc)
https://www.rsscaz.com/ (rio)
http://www.desertsharpshooters.com/ (avery)
https://www.facebook.com/AZLRPRS (cowtown)
https://www.roadrunnerrifleclub.com/ (avery or rio)
http://arizona-rifleshooting.com/
AZ has open carry and constitutional carry. Take the AZ CCW class anyway, its useful info and the permit is your proof of background check so you don't have to go thru NICS when you buy.
There are 4 or 5 nice outdoor shooting ranges in the Phoenix area. See below. (As well as several indoor ranges I don't know much about.)
I've been a member of both PRGC & RSSC for over 20 years, (and have been a volunteer, match director, coach or club officer in those years).
Both RSSC & PRGC are true member-run clubs, both lease their dirt (RSSC from AZGFD, PRGC from PHX city park).
Ben Avery is a AZGFD owned and run facility, landless clubs reserve the various ranges and run shoots there.
A smaller range, Joe Foss range in Buckeye, an hour west of Phoenix, is run by ASRPA.
All 3 larger ranges have a public range, the 2 clubs offer membership. All 3 have lots of competition style events. Every scheduled competition or practice style event at the 2 clubs and the clubs that use Avery are open to everyone, most by walk-on, some shoots take advanced entry reservations.
Both PRGC and RSSC cost about the same to join. Each has a main range and 6 or 8 specialty ranges. RSSC faces north, PRGC faces south (low winter sun is unpleasant to shoot in) PRGC High Power range is configured properly with 1 pits and 3 firing lines, Rio is backwards, 1 firing line and 2 pits.
RSSC tries to run itself like a business, all members have to leave when the last RO clocks out, even if there are still 3 hours of sunlight remaining. All the specialty ranges (except shotgun) require a lengthy approval process to gain access. On top of that, there is a tedious process to get the after-hours pass that allows weekday shooting till 9. RSSC has the best sporting clays field in the state, perhaps all the states. But they don't allow carry.
PRGC is much more member-friendly. members can use most (but not all) ranges without a special qual process. PRGC is very friendly for members bringing guests and allows carry.
BASF has about 80 firing points on the public range and 15 or 20 trap/skeet/clays fields and a huge archery area. The facility also has about 15 other ranges reserved by clubs for shoots including 100 firing points to 1000yds on high power, 2 silhouette ranges and about a dozen practical bays.
PRS-style shooting is monthly at a private shooting site called "Cowtown" in the west part of the valley not far from Ben Avery. And a monthly shoot at RSSC.
BLM is opening some formal maintained shooting sites in the west valley desert.
There are some nice outdoor ranges in Tucson, also near Casa Grande, Flagstaff, Payson, Kingman, Yuma and other parts of the state.
As to shooting in the Desert, BLM and National forest permit shooting most of the year. State Trust Land permits hunting but NOT target shooting. Indian reservations do not permit anything. Check the national forest websites, Tonto NF, the one closest to phoenix, has closed off shooting on hundreds of square miles nearest town due to dirt bike infestation and bans water jugs. Steel and paper are still OK, but plan on an hour drive to find a safe legal place. During hot weather fire seasons all the NFs ban shooting, camp fires and smoking.
Hunting is by drawing for tags to hunt anything larger than a quail or coyote. But lots of public land to hunt on.
https://www.azgfd.com/
Shopping in the Phoenix area:
BassPro in Mesa
Cabela's in northwest valley
2 or 3 Sportsman's Warehouse stores
lots of Local gun Shops.
Hope this helps, welcome to AZ, shoot good!
Poole
http://arizona-rifleshooting.com/
more:
The problem with "rural" in AZ is nearly all the land is BLM, national forest, state trust, indian rez. (except for rez, all are open to hunting with an AZ license, BLM and NF are open to target shooting, except for the crowded areas infested with dirtbikes and the entire state during fire season (every summer!), so "rural" is far away or expensive, often both. if not there may not be water. I'm speculating, but I bet there are probably few places in OH where a 50' or 100' well does not produce water, there are fewer places in AZ where a 100' well is deep enough or any well at all is even possible.
"Phoenix", the area, is the city of Phoenix and the 20 or so cities and areas that form its suburbs. Sometimes called the "Valley of the Sun" (in summerr its more like the "Surface of the Sun" :) but we like it this way!
There are some rural-ish locations around Phoenix. South: Maricopa, Casa Grande. SE: Apache junction, Florence, NE: Cave Creek/Carefree. North: New River, SW: buckeye & beyond, and NW: beyond sun city out to Wickenburg. All of these put you about 1 to 2 hours from PHX airport.