I just moved from Phoenix so I'll try my best to help. Phoenix is a great town, but it is definitely hot as hell. Be prepared for over 100 for a good portion of the year. Unfortunately, it really doesn't cool off at night, so it's pretty hot even in the evenings and early morning. It all depends on your level of heat tolerance. Still, most gyms are good about having plenty of A/C.
It really depends on what kind of gym you are looking for. If you just want a commercial gym then there is Gold's Gym by the stadium in Peoria that is good. A few Bodybuilding pros like Troy Alves train there, so they should be pretty forgiving about heavy lifting. http://www.goldsgym.com/gyms/index.php?gymID=961
If you're looking for a real powerlifting gym, there's Raging Inferno Power (R.I.P). It's out in Peoria and it's the real deal. Don't go there unless you plan on competing in Powerlifting. Here is some info so you can get in touch with them:
Die Hard Gym in Peoria is another good one. They host powerlifting meets all the time, so they would be a good resource. http://www.diehardgym.com/ Might be the best of both worlds; very PL friendly but not totally focused on competition.
For God's sake don't go to any Pure Fitness or LA Fitness.
Let me know if you need any more info. Hope this is helpful for you and welcome to the desert!