You definitely want to climb Elk Mountain... close to the city center, little traffic (alot of bike traffic though!), incredible views, challenging, a great way to get up to the parkway (loop back to craven gap, come down town mountain road if you need to get back into town... go north to craggy if you want to do some serious climbing and see one of the best views around!)
The parkway is excellent - either south toward pisgah or north up to craggy gardens.
Along the french broad river to Marshall is a really nice ride, about as flat as it gets around here but quite beautiful.
Asheville Cycling Road