Right from town there are easy paved biking paths.
On the east side of the town along 191, there is a path that leads north towards the Colorado River.
From this path you can cross the river and head up towards Arches and the Bar M trails or turn right at the Colorado river and ride along the river towards the end of the Porcupine Rim trail.
The ride across the river, up the Moab Canyon Pathway takes you right past Arches and then climbs up to Bar M.