I hiked to Delicate Arch before sunrise one morning, about 3.8 miles round trip. It was great to sit there and watch the sunrise. There were a few other people there but it was not crowed. You can walk right down under the arch if you wish.
Ezpeletas B & B raises goats. They take them for a walk up on the red rocks rising from their parking lot.
Near Monument Valley it's open range. Here were a couple of horses near the road with the valley in the background.