North of Moab is one of the most scenic roads in America. Locals call it "The River Road". The river road drives along the Colorado River in the bottom of the canyon. It's like driving in the bottom of the Grand Canyon (almost). If you are headed to Grand Junction on your way out of Moab, the river road will take you to Interstate 70, and only add 10-15 minutes to your drive.
If you aren't headed to Colorado on your way, just drive up the River Road 15ish miles then turn around and head back. Dewey's Bridge is a good turn around point.
This link will take you to Dewey's Bridge.
The bridge burned down years ago... there are several versions of the story depending on who you ask locally.
On this trip you can also stop at the FREE Movie Museum located at the Red Cliffs Lodge.
Directions to Movie Museum