I'd second KLR's recommendation of the Jailhouse Cafe for a great breakfast. There's also a decidedly non-gourmet brewpub that has some very nice beers and good enough food.

A couple of excursions that I'd recommend, depending on how long you're going to be in the Moab area, are Little Wild Horse / Bell Canyons and Horseshoe Canyon. Each is a day trip in its own right with a couple of hours of driving and a half-day of hiking involved, but they're both fantastic experiences, and get you a lot farther away from the pavement than the typical day in Arches will let you.