You don't have where you live. We have taken 20-some years of horse riding/family vacations (also camping with the dog) principally bc it was cheaper to taking riding vacations that to go somewhere else and pay for the animal sitting.
I would start by limiting myself to one section of one state, unless you have 3 months of vacation time. In CO, where my DH's grandmother grew up and he has to go each year for his "fix", you'll be hard pressed to find any riding just west, northwest and southwest of Denver. There is a BIG, BIG suburban Denver sprawl, which you can't see until you drive around, and it's very limited to any places that will let you ride your horse. Take MY word, it's really over-developed.
I would go on the I'net and look for privately owned horse camps further west, or SW, like around Durango, and avoid the high peaks. Talk to the owners and see what is available. Sometimes there are miles and miles, and sometimes there are 3 rides total you can take. It is a 20 hr drive from home to Denver for ME, and I don't want to travel that far for a limited few trails. In 1987 we took a small group out and rode from Empire on a very old mining road/trail, up to Waldorf. On one section of the trail, it was SO overgrown with aspen that we sent our herd leader through on foot, and then sent each horse through one at a time. I'm sure it is IMPASSABLE today, even for a hiker IF you could find it.
It sounds like Utah has some nice places to ride, but I've never ridden there. Put it on my list!
There is good riding in the Black Hills, SD. It feels like CO, but the dropoffs aren't as treacherous. We've stayed in several camps there, but I would recommend starting in the French Creek Horse Camp, in Custer State Park. It's a little more pricey than other places, but every ride takes you into the park, where you can ride on the paved or gravel roads, or ride the trails. Leave the buffalo alone and they won't bother you. Also, get your reservation before January 1st bc they fill up fast. Don't bother camping anywhere that juts up to the Mickelson Trail. There are so many "don't do this" and "don't do that", that DH and I decided we wouldn't.
We're going back to: Elk Haven Horse Camp
There are only a few rides that originate from the property, but we're planning to to trailer out for the rest of them.
You WILL have to buy and feed certified weed free hay. At a privately owned camp, like above, you can feed your own, but you can buy some to take with you on rides in the park or in The Black Hills National Forest areas. They carry that for sale, too. The same is true with EVERY state park or national forest out west now.
Hope this helps you plan. =D