I've spent 70-80 days running the trails around Moab. There is tons of stuff you can run ... I'd recommend some rocksliders. But, they're not 'necessary' if you stay on the milder trails.
Below are two lists of trails and their ratings. Ratings are subjective, though - you'll notice the two sites rate some trails differently. So, take the ratings as a general guide, not gospel ... and ask around.
Cruise Moab Trail Ratings
Red Rock 4-Wheelers Trail Ratings, EJS 2012