Good luck with free camping, even if you drive out of town to the BLM sites you still have to shell out $5-$10 a night. Last September I stayed a week at Up the Creek Campground
. It's 15 bucks a night for a single person, less per person for groups. Nice central location right in the heart of Moab. Very nice sites and probably not crowded this time of year. It includes hot showers, picnic tables, green grass, and a little creek running right next to the campground. I haven't done much hiking but check out the Porcupine Rim, Bartlett Wash, and Gold Bar/Portal trails for mountain biking.