The area around Moab is filled with hiking trails. You can find many of them using the ALL Trails application for your phone. Many are remote without cell service, so prepare yourselves well and download maps to your all trails app. Some of my favorite trails are listed below, with links to the trailhead in Google Maps. Read the Reviews on Google for more trail info, view the Google Pictures, then click "Directions to get you to the trailhead.
Corona Arch Trailhead Nice hike near town with ladder and easy handholds for a unique trail.
Fisher Towers Trailhead Hike at the base of 1000' towers with a couple of ladder features.
Grandstaff Canyon Trailhead Follow and cross the creek to Morning Glory Arch at the end.
Hidden Valley Trailhead Hike to a hidden valley in the cliffs south west of town.
Mill Canyon Dinosaur Bone Trail Hike the trail and see dinosaur bones in the cliff face. Past the footprint location.