Ten or twenty years ago, there were many horse corrals all across the Southwest that rented riding horses and pack horses to hunters, fishermen and wilderness campers. You could go to one of those outfitters and rent a couple of horses for a week, then set off into the wilderness on your own. Today, there are far fewer establishments that would rent a horse for such a long period.
There are still many outfitters in the Southwest. Here is a list of outfitters in Utah
Most of the outfitters now rent horses for day trips along with a guide. They can make more money offering guided horseback trips to the tourists than they can renting a horse. It can be a big risk to allow a person unfamiliar with the country to take a valuable horse into the wilderness. Some of these outfitters will organize guided pack trips that last a week or more, but that can be rather expensive.
Dude ranch vacations are not cheap, but the price normally includes accommodations, meals and the use of a horse for the entire week. When you consider all that the price may be a bargain. I suggest you look at the Dude Ranches listed on our Dude Ranch Page, and find one that is less tourist oriented and more horseback oriented.