There's a tremendous amount of stuff right outside of town, literally a stone's throw.
Pick up the Latitude 40 East and West maps, and do some Googling on the area. You'll come up with lots of trails based on the kind of stuff you'd like to do - then find it on the Latitude 40 maps.
In addition to the Latitude 40 maps, which are of excellent quality, this book is pretty useful:
http://www.amazon.com/ATV-Trails-Guide-Moab-UT/dp/0966497678It gives a great overview of some good routes in the area.
The Latitude 40 maps are available at the Moab Visitors' Centre - and it has tourist friendly hours

We were astonished to be able to buy our maps there on a Sunday afternoon.
Enjoy Moab. We're planning to be back again in March.
