You are not far from Main Street Billiards, on Main Street in Mesa, East of Dobson. Great tables, food and plenty to drink and there is always something going on. I'm there daily, action every now and then and always someone will to play just to play.
Kolby's as well, on University, West of McClintock in Tempe. Seems the crowd is small lately, but there is a lot of one pocket action by the regulars.
You also have Jester's Billiards, Baseline and Gilbert, however, that seems to me more of a "banger"/College crowd and honestly, the tables aren't exactly great. I was in there a few months back and there wasn't much happening and the 3-4 tables I tried out were horrible.
Shoot me a PM, maybe we can get together sometime. If you're on FaceBook, find AZ Pool Scene and "like" that page, there is a lot of good info there with tourney postings as well.