Ar_Clint - Man you did not get much of a responce to your question. There are quite a few Phoenix area hunters on the board, I'm not overly familier with the Phoenix area but here's my best shot at information for you.
Sure there are lots of Varmints to hunt in Az. For Prairie Dogs you are going to have to head up to the Northern part of the state. Seligman is the most talked about location but they are scattered all along highway 40. Arizona PD's are not exactly what I'd call classic PD huntin'. There are sometimes huntable numbers in Az but it can be spotty. Verious diseases and low rain fall has taken a hit on the PD populations in the past 10 years. I've hunted them in Seligman, north of Williams heading toward the Grand Canyon, in Wynonna (SP) eaast of Flagstaff (I've heard that this area is closed to PD shooting now). Lots of people hunt them on the 4 corners Indian Reservation and down near St. David. You need to consult your Reg.s because there is a season on them here. Oh by the way, New Mexico has some PD's as well.
Other varmints would include Black Tailed and Antelope Jackrabbits. I don't have any idea where you'd hunt them in the Phoenis area. Again consult your Reg.s. You are in one of the Metropolitan areas and they have special weapon laws there.
If you can find a large colony of Round Tailed ground squirrels you can have some mighty fine shooting with a 22 lr. These things are much the ame as hooting PD's but smaller. Great fun with a 22 lr.
If I were you I'd join one of the calling clubs up there. There are two very active clubs in the Phoenix area. Those fellas can get you going in a hurry.