Okay, I will tell you about my favorite forest service road, 475B, which is actually south of Asheville in the Pisgah National Forest near Brevard. Slick Rock Falls, the Fish Hatchery, Daniels Ridge, and Looking Glass Rock trails are all located off this service road. You just pull over and find a sweet spot to camp! No facilities and very primitive, but my favorite!!
There's a small charge for Lake Powhatan Campground in Bent Creek Area in Asheville, but it has GREAT trails, near the botanical gardens, is in a beautiful forest, and has a fun little lake with a sandy beach. This is definitely family friendly.