There are a variety of species that are commonly hunted in Utah. These include mule deer, elk, antelope, pheasant, ducks, geese, sage grouse, forest grouse, chukar partridge, mourning dove, and wild turkeys. Before you embark on your trip, make sure that you obtain a Utah hunting license.
Information can be found at http://wildlife.utah.gov/index.php
For the best experience, you will also want to choose from one of the many Utah hunting guides. Here are just a few of them.