Among your first stops should be Island in the Sky Visitor Center. Here, you'll be able to orient yourself and pick up some great ideas before you start exploring.
Wandering around this tourist information center won't cost a thing, which is perfect if you're chasing free things to do. You'll find Island in the Sky Visitor Center in Moab. There are plenty more attractions to discover in the vicinity, including Dead Horse Point State Park, Mesa Arch Trailhead and Neck Spring Trail Trailhead.