well.. i live north of asheville.. about 30 miles north.. but honestly i have not found any GOOD trials in the area unless you go down to Murphy, NC or to the Brown Mountain ORV area..
There are a lot of forest roads and such that lead into the Pisgah National Forest.. they are everywhere.. some have gates that are closed during the off season.. im sure they would be open in springtime.. there are some like in my area that are remaining closed because of damage from hurricanes few years ago flooded the area. I am not sure about asheville. There is always the scenic route of taking the Blue Ridge Parkway.. my favorite pastime.. i take it to goto walmart even
If you like hiking.. you probably know about Mt. Mitchell and the Black Mountains area.. there are TONS of hiking trails.. from beginner to extreme.. in fact the Colberts Ridge Trail leads up to it like a minute from my house. You have to take the Blue Ridge Parkway north from Asheville to get to Mt. Mitchell.. the trip is awesome..
http://www.northcarolinaoutdoors.com/places/mountains/blacks.html
http://www.ils.unc.edu/parkproject/webcam/webcam.html