The problem with greyhounds is that few of them are safe around cats - they’re gentle, sweet dogs though.
I also ditto the suggestion to find a dog that’s already been fostered in a home. The foster “parent” will have very good insight into what the dog’s general temperament and traits are - something you’re not likely to get from the shelter. The up-front cost of getting such a dog would be worth it - look at it this way, most shelters are supported by taxes and licensing fees so one way or another you’re going to foot the bill. $200 is cheap for a speutered, vaccinated, heartworm-free dog. Amortize that number by the number of years you’ll have left with the dog. Twenty bucks a year? 
I’ve been involved and fostered dogs for almost 20 years - in most cases, what the rescue charges for dogs does’t even come close to covering actual expenses. It just offsets them a bit. Most people involved put a lot of their own time and money into the dogs (and cats) in their care.
Another option would be to call and see if your local shelters have a dog-walking or fostering program, so that you can “try out” different dogs.
You and your son sound like you’ll be great dog owners. I’d strongly suggest that you arrange to take your new dog
to basic obedience classes, just for fun and so everyone learns. Money may not be a primary concern, but you do need to be aware that you’ll have to spend money on flea and tick prevention, heartworm preventative (don’t know how much of a concern that is in Tucson), as well as related costs like food, toys, crate, licensing, classes (if you so choose) and the general paraphernalia that a dog needs as well as occasional veterinary attention. The smaller the dog, the cheaper, typically.
One more quick tip: Research breeds. Most (purebred) dogs were developed for a purpose. Some are more likely to need a lot of exercise, be vocal, dig for various reasons, have high prey drive, be aloof or territorial, etc. Not every dog of that breed/mix will have those traits, but getting, say, a beagle or beagle mix and expecting it to be quiet and OK with lots of time alone (like most hounds, they were selectively bred for voice and to work in a pack) would be unrealistic.
Three, if your son likes to compete but isn’t crazy about team sports, dog obedience rallies, agility, flyball etc are a great way to get out there and compete. Agility is fun and fast paced, great for an active dog and an active kid.