The Dinosaur National Monument is in the extreme northwestern corner of Colorado and the northeastern corner of Utah.
Highway 139 (plus changing to highway 64 at Rangely, CO) north out of Grand Junction, CO is a very scenic route to get to Dinosaur. Douglas Pass is a seven-mile stretch of incredible mountains and very winding roads going up one side of the mountains and down the other side. Plenty of photo ops along the way through it. It takes about two hours to go the 109 miles between Grand Junction and Dinosaur, CO. There are several roadside signs along the way declaring that the area is "open range" (ie. the cows go anywhere they want). There are several roadside signs denoting a "cattle gate" ahead.
Alternate routes are from either
Once you reach Dinosaur, CO at the junction of highways 64 and 40, you have two choices.
The quarry in Utah shows many dinosaur bones still embedded into the unquarried rock.
(1) Wyoming (Rock Springs or Green River, WY going past Flaming Gorge National Reserve) to Vernal, UT at the junction of highways 40 and 191, or,
(2) Utah (Salt Lake City or Provo, UT) to Vernal.
In fact, unless you want to double-back for several hours of driving, it is suggested that you come in one way and leave the other way.
(1) Go east on highway 40 one mile to the Colorado visitors center. From there, you can take the 20-plus route into the Colorado side of the Dinosaur National Monument. No dinosaurs to see, just great scenery.
(2) Go west 20 miles on highway 40 to Jensen, UT. Again, spectacular scenery along the way. From there, you go north 7 miles to the Utah visitor center. This is where the quarry and gift shop are located. A tour guide will explain much of the history of the archeological dig. You can also self-tour the two-story facility.