There are more options than it is really possible to list and if you search I think you will find out this question has been addressed a ton on this forum...
that being said one spot I really like is up Ute Trail. It is the north side of the valley. Turn over the river at Sand Lake, following signs to Spiral Drive, don't turn in Spiral Drive and start driving up into the hills. The road will turn to dirt and after about 15minutes or so you will come to a Y. Go right and there are a bunch of nice places to camp. I don't know all the County road names but you'll figure it out. Otherwise there are millions of acres of public land around town. Drive into the forest in any direction and you will find somewhere to camp. Have Fun!