My suggestion would be to (Google) contact a bike shop there in Moab. I would think they have mountain bikes to rent also? Have to check on that though. Once your in the Parks there is no end to the dirt and some pretty awsome views.
San Juan County is basically all two lane roads with speeds roughly 65 MPH. You could ride from Moab to Monticello and back or go one way. You will see a lot of Trucks, like Ford FXXX and other makes. The roads look "country" but are lifelines between these cities that are so far apart. It would not suprise me if people are going faster than the speed limit. I would definatley ask the bike shop(s) on this.
I hope this helps. Get some more information for sure.
Red Rock