Wyoming is my favorite state in the West to hunt. It's very sparsely populated, has tons of public land, and has healthy, diverse wildlife populations.
Antelope are the signature species of Wyoming, which has by far the largest herd in the nation. I've hunted antelope in WY every year but one since 1994, and look forward to my hunt there every year.
Wyoming has good mule deer hunting throughout the state, and good whitetail hunting in the east. Wyoming Game and Fish does a nice job in managing the elk herd for quality.
Wyoming has healthy populations of premier trophy animals such as sheep and moose. They also offer wild bison hunts, one of the very few states to do so.
Wyoming does have a small population of mountain goats as well. The opportunity is limited, but good goats are available if you're lucky enough to beat the odds and draw.