Start off by calling Ben Avery, also in Mesa call over to Caswells Shooting Range and Ted's Shooting Range in Queen Creek to see what matches they might run. North Phoenix look at Scottsdale Gun Club for information.
There is an Arizona specific firearms forum, http://www.arizonafirearmsforum.com/, a bunch of very good men and women.
I'm a member of the Phoenix Rod & Gun Club. Our club range is right next to the Phoenix Police Training Academy. We have several rifle, and pistol ranges out to 500 yards. They have A lot of IDPA, as well as other forms of competition like 3-Gun and High Power Rifle. It's a nice place to shoot, with a lot of members that are very informal and friendly. Here are a few videos of the matches they have going on there.