Stayed in Green River for a week in 2000 to study rocks and map sedimentary structures in the book cliffs. The area was an analog of the field we would develop over the next two years. The area is used extensively for field training geologists because of access to the same formations at many different points over a large area. If you are not a geologist, I guess bare rocks are scenery?
That trip, we used a motel, you don't really "camp" with a group of 20 geologist and engineers for field work unless there are no other options.
For what we were doing, going up oil field service roads and following powerline and pipeline inspection routes for access, 4wd was necessary to get to the places where we would start hiking and climbing. There are a few places where you can get close to the rocks with a car, 4wd takes you further, dirt bike or small 4-wheelers would be even better, but you'll still put in a lot of miles on foot to really see the area.
In 2006 I was there for an overnight in the RV. Stayed at Shady Acres RV Park, which was adequate; restaurant within walking distance, and at least one tree, we parked next to it.
Trees are pretty special in that part of Utah. Other than a stop at the park service museum/visitor center, and buying melons, we didn't do much in Green River that trip, we were headed toward the canyonlands.
Downtown Green River on a Sunday morning early in October:
If you are looking for "developed" recreational and tourist facilities, you'll find more of that down the road in Moab, but you are in different rocks that tell a different story.