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.
When you've spent a bit of time at this tourist information center in Moab, why not check out Dead Horse Point State Park, Mesa Arch Trailhead, and Neck Spring Trail Trailhead? They're not very far from Island in the Sky Visitor Center and also a hit with travelers.