About 145-200 million years ago, when Utah's climate was wet and warm, dinosaurs laid down footprints (and their dead bodies) in sand and mud. These traces were covered and hardened under the pressure of overlying sediments. As mountains uplifted, the sedimentary layers were propped up and exposed to erosion, eventually revealing the fossilized bones and footprints.
The Museum of Natural History  is pretty darn good, even for the jaded fossil-watcher.
The Utah Field House of Natural History State Park and Museum (http://stateparks.utah.gov/parks/field-house ) has an interesting introductory video and a great collection of dinosaur bones and fossilized plants and mammals from the Eocene Epoch.
Head north of town to the northern edge of Red Fleet State Park (www.utah.com/stateparks/red_fleet.htm ) and take a three-mile round-trip hike to see dinosaur tracks.
Visit the College of Eastern Utah Prehistoric Museum (www.ceu.edu/museum ) in Price to see locally dug fossils, including the Utahraptor. (Remember Jurassic Park?)
After the museum, drive east from Huntington to the Cleveland-Lloyd Dinosaur Quarry (www.blm.gov/ut/st/en/fo/price/recreation/quarry.html ) and visit the excavation site.
The Museum of the San Rafael (www.castlecountry.com/what_to_do/san_rafael_museum ) has more bones and a replica of a fossilized egg. Look for dino tracks as you explore the San Rafael Swell.
The St. George Dinosaur Discovery Site  at Johnson Farm is considered one of the world's best dinosaur-tracks sites. Recently discovered tracks show early Jurassic dinosaurs absolutely running amok across the former lakebeds here.
The Museum of Ancient Life (www.thanksgivingpoint.org/visit/museum_of_ancient_life/about.html ), just off I-15, has a huge collection of dinosaurs and other fossils and shows IMAX movies with computer-generated dinosaurs.
The Big Water Visitor Center  on Highway 89, at the southern edge of Grand Staircase-Escalante National Monument , is built in a spiral to resemble a fossil (not a dino, but an ammonite). Exhibits focus on the monument's paleontology.
The Museum of Moab  has good exhibits, both of dinosaurs and local history. Trails outside town lead to tracks of a theropod and meat-eating sauropods.