When I lived in Gilbert, I planted almost anything and everything. First, make sure to mix lots of rich compost into your soil, as the rock-hard clay soil will not grow veggies without some help from compost. Better yet, build a raised bed and fill it with garden soil.
There are two growing seasons there, one in the fall and another in the spring. The spring garden is the best, and I usually planted in early to mid February. Look for veggies that have a shorter "days to harvest" listed on the seed packet, as you will start to run into hot days by the end of May.
If planting in the fall, I usually planted in late Sept or early Oct, because sometimes it gets too cold in Dec/Jan and there will be a freeze. Also, it is helpful to have some sort of removable shade cover for your beds for when it gets hot. I built one using some shade screen (from home depot) with grommets in the corners and stakes to hold it up. That way I could leave it off in the morning/late afternoon so the plants could get sun, but I put it on during the hottest part of the day.
I also grew a grapefruit tree (the best!) and a peach tree. There are certain varieties of fruit that grow best there, so you will want to check with the nursery on that. For example, my peach tree blossomed in January and produced fruit in May. Happy gardening!