I live in the desert and want to grow fruit in my garden but am concerned about the heat.
I'm wondering what varieties the community would recommend I take a stab at when it begins to warm up?
For perennials, dates and figs might work, as well as all sorts of cacti (you could do a whole garden with them, since there are loads of kinds). Both the pads and the fruit are edible of most Prickly Pear cactus varieties.
Similar to Debbie M.'s answer, most things in the Prunus genus may work well: almonds, apricots, peaches, nectarines, plums, and sour cherries.
I wouldn't recommend apples, particularly (they're not in the prunus genus, either), but apples should work better if they have good ground cover (I'm still not sure how they'd do where you are, though).
I have a relative in southwestern Utah (by Arizona and Nevada) who grows almonds and figs. It's pretty hot and dry there. They seem to be quite fruitful.
For annuals and tender perennials, try these: Red-seeded Citron watermelon, Kiwano, Dark Star zucchini, Beit Alpha cucumber, Suyo Long cucumber, Camp Joy tomatoes (sets fruit up to 115° F., I've read), Punta Banda tomatoes, Taos tomatoes, some of the tomatoes at goodmindseeds.org, etc. Kiwano and Citron watermelon grow in the Kalahari desert. You might consider New Mexican landrace peppers (they won't have New Mexican in the name, however, due to some weird laws that restrict that to certain other peppers).
See this question/answer for more information about heat-tolerant tomatoes.
If you have clay or clay loam soil, I recommend taking advantage of it. Slow-draining soil that holds moisture well can help in the desert (and you shouldn't have to worry about root rot, because of the aridity and such). If it drains quickly, on the other hand, cacti might be your best option, since they like that, I hear. Peppers usually aren't too fond of clay, though (but some like it more than others). Tomatoes can handle clay well. So can zucchini.
Search for drought-tolerant varieties, too. Tabasco peppers are very drought tolerant. Same with Punta Banda tomatoes. Drought tolerance is just as important as heat-tolerance in hot/dry areas.
Potassium is supposed to increase heat-tolerance. So, make sure your soil has enough. Also, make sure it's not magnesium deficient, since that could cause problems with heat, too.
As long as you are willing to water there are many options. Fruit trees that will work: low chill, early ripening apples, peaches, plums, figs, pomegranates, nectarines and apricots and virtually all varieties of citrus trees.
Grapes can also be grown, as well as melons and strawberries.
I have seen various bare root berries available (my two attempts at raspberries failed) but I've not really heard of anyone growing them successfully.
Your local garden center should carry a selection of plants that can be grown in your area.
Dragonfruit (pitya) is a cactus that bears large edible fruit. Other things to consider would be Pomegranates, pistachios, maybe even date palms... Probably lots of normal stuff "can" survive there with a little help, I would talk to someone at a nursery in or near you area (maybe Phoenix or Scottsdale)