Jeff,
I’ve hiked with dogs for about 2500 miles in Arizona and Nevada so I might have experience that’s relevant to you in Utah.
Hiking with dogs is my absolute favorite way to spend my time. If I could, that’s how I would spend every day of the rest of my life. And I’ve come to learn that dogs are like kids – the more time that you spend with them, the better they turn out. So the time spent hiking together will increase your appreciation for them. As Randy said, if you love dogs and you love backpacking, you’ll love backpacking with dogs. Look at my boy sitting in his Helinox:

You should be aware that a dog’s individual personality has a great deal more effect than does the breed. I’ve had dogs that loved covering big miles, dogs that tolerated it, and even one that was, like Jim Gaffigan, more “indooorsy” and ran in the opposite direction when she saw me with a backpack. So it is possible to get an athletic breed but a dog that just doesn’t enjoy hiking.
I encourage you to rescue from a shelter. Rescues are grateful and so they bond more closely to you. I’ve had a number of dogs both from breeders and from shelters, and the difference has been quite noticeable. I’ve loved them all but the rescues have been the best companions. If Utah is like Nevada, you will find cattle dogs at the local shelter because ranch dogs are rarely neutered.
Hiking in the desert with dogs presents challenges. The anatomy of a dog makes them more susceptible to overheating. Their bellies are like sails catching the heat radiated up from the ground. So winter is much better than summer (plus, no rattlesnakes in the winter). I carry a lot of extra water in the summer to pour over my dogs and to let them drink frequently. In the winter, they hardly drink at all because they eat snow as they hike.
You’ll be amazed at how they move in the mountains. The first time I took my dogs from North Carolina to Nevada, I was nervous about hiking in the mountains. My girls were just two Southern belles from the Carolinas that had only experienced the grass backyard of my place outside of Ft. Bragg whereas I grew up in the Nevadan mountains and had served in the Army’s high-altitude special operations unit. But they smoked me. I couldn’t keep up. They were so sure-footed. By the third day, I realized that I was seeing the advantage of being a quadruped. It’s not just deer and sheep that can fly up a mountain. As has been said in this thread, a harness is very useful in the mountains. I use a military working dog harness so that I have both a handle and a rope attachment point. I used to have the dogs carry some stuff in side pockets (a leash and contact info in one and a fleece and poop bags in the other) as you can see here:

But I ended up removing the side pockets when I saw how difficult they made navigating between large boulders. The dogs don’t seem able to grasp that the pockets made the dogs wider than normal.
As far as pad-protection goes, I also like dogbooties.com. I found Musher’s Secret to be of no value in the mountains (to be fair, that isn’t the intended use for the product – it’s to keep ice from between their toes). I also tried the more expensive vibram-soled booties but could never get them to fit properly. The cordura dog booties are cheap and make a large difference in how quickly a a dog will move over sharp rock:

Hope this is helpful.
Kris