@fiorite- Finding fossils in the desert of Utah could be especially difficult unless you are knowledgeable of the local geology and rock formations. If you are going on a trip to Utah, it might be a good idea to find a fossil quarry. For about $50 you can search for fossils in privately owned fossil beds. You will not find fossilized dinosaur bones, but you will likely find trilobites and other marine invertebrates.
These fossil farms are usually made up of limestone shale that splits easily into layers, revealing fossilized organisms. I spent four hours at one of these fossil quarries with my daughter and we found nine trilobites, a sponge and a partial phyllocarid (prehistoric shrimp) in all. One of the trilobite fossils we found was really nice and about 1.5 inches long.