Bicycles must stick to established roads in Arches National Park [1]; cyclists have to contend with heavy traffic on the narrow paved roads and dusty, washboarded surfaces on the dirt roads. Beware of the deep sand on the four-wheel-drive roads. Nearby, BLM and Canyonlands National Park [2] areas offer much better mountain biking.
Rock climbers don't need a permit in Arches, although they should first discuss their plans with a ranger. Most features named on U.S. Geological Survey maps are closed to climbing. That means any of the arches and many of the most distinctive towers are off limits. There are still plenty of long-standing routes for advanced climbers to enjoy, although the rock in Arches is sandier and softer than in other areas around Moab.
Several climbing restrictions are in place, however. No new permanent climbing hardware may be installed in any fixed location. If an existing bolt or other hardware item is unsafe, it may be replaced. This effectively limits all technical climbing to existing routes or new routes not requiring placement of fixed anchors.
Links:
[1] http://www.moon.com/destinations/utah/canyonlands/arches-national-park
[2] http://www.moon.com/destinations/utah/canyonlands/canyonlands-national-park