The main sites are reached via a short loop road that is suitable for bicycles as well as cars; in fact, it's quite a comfortable ride that gives a more intimate sense of the canyon than you'll get from a car. If you are biking, make sure your tires resist punctures, as all manner of plants with thorns and spines grow in the park.
The short trails to the sites along the loop road can be visited in street shoes, but hiking boots are a good idea if you're planning on visiting any of the more remote areas described under "Do." Some but not all of the sites are wheelchair-accessible, as is one of the campsites at the park campground.
