hmmm... good question. I know there are some small ruins nearby in Canyonlands and plenty of pictographs in the Moab area but not sure of their access unfortunately.
Your best bet if you really want to see something spectacular is drive 2 hours south to either
Hovenweep National Monument or the Grand Gulch area west of Blanding near Natural Bridges National Monument. Mule Canyon has some very accessible ruins if I remember correctly.
I would suggest asking the helpful folks at the visitors center on Main Street in Moab if they have any suggestions. If you visit something nice please post it after your trip here for others.
Chris