With a good used bicycle, you will get a lot more for your money, if you know what to look for, can do some maintenance, and can find one that fits. Going to a store makes it more likely that you'll have a bicycle that fits you well--very important--and doesn't need any maintenance right away, but you'll get a lot less for your money.

A non-suspension mountain bike should do well, especially if you put smooth tires on it. Personally, I hate flat bars, and would switch them, but that would cost more money, and you may be happy with flat bars.

You definitely should consider not only a rear rack, but also fenders. It's a good idea to carry a pump and a few tools, just in case you need them. Lights can be very helpful, and are essential if you think you may ever end up riding in the dark or in low light-conditions--I like generator lights, but you can get decent battery lights for a bit less than the cost of a generator system. Warm clothing is pretty important here in Ohio, but you may not need it in Arizona.