This Saturday (5/21) and Sunday (5/22) from 7:30am - 4pm, is the last weekend of the Peach Festival out at Schnepf Farms, located at 24810 E. Rittenhouse Rd. Queen Creek, AZ 85142. Along with picking peaches, apricots and other fruits, there is food sampling, celebrity chef demonstrations, carnival rides, tractor rides, a petting zoo, pie eating and even peach pit-spitting contests. Prices do vary for activities and food, such as pancakes, burgers, hot dogs and chicken, but admission to the festival itself is free. The article by the East Valley Tribune states that the fruit is $1.90/lb.
While this is the last weekend for the festival, you can pick peaches, apricots and other fruits every Thursday through Sunday in May from 7:30am - 4pm, which leaves next Thursday - Sunday (5/26 - 5/29).
If you're interested in pricing and the time of different activities please check the event calendar at the Schnepf Farms website.