I live and ride on the other side of the Blue Ridge Parkway from Waynesville. I've never ridden my bike over there but have driven through it a couple of times. Bringing your bikes is dependent upon your sense of adventure. We don't have hills here, we have climbs. I average at least 2000 ft of climbing on an easy ride. This is not meant to scare you, just inform. Because my partner is also a cyclist - probably in the same fitness as you - and she finds predominantly flat roads to ride 3 days a week. There are also several bike paths around the general area if don't mind driving to ride. (I do, so I ride what the road gives me) The most known is the Swamp Rabbit Trail that runs about 20 miles from the NC border (around Traveler's Rest) south to Greenville, SC. A great place to spend some time, too. It also looks like there are roads that circumnavigate Lake Junaluska in Waynesville, so that should be pretty flat and scenic. I'd buy a local road map, find an interesting route and then drive it first. No nasty surprises, much more fun.

For 3 weeks, bring your bikes, you'll find places to ride and for the most part feel safe and comfortable. The roads are narrow and winding but most drivers are courteous - sometimes too much so. Don't let the mtn bike hype turn you off, this was road biking country before there were mtn bikes. But WNC is quietly building its reputation as mtn biking mecca, so its hard to avoid the subject. I ride the woods in the winter, its less crowded and warmer than the road.