There was a recent discussion of bikes on this thread:
//www.city-data.com/forum/ralei...cary-good.html
As I posted in that thread, I agree with Dire Wolf that road biking in this area can be a little hard. Many people do group rides for safety more than other reasons; my sister road bikes a lot but strongly prefers a group. Because we have small children we stick to trails.
The best places for road biking are probably the state-sanctioned ones you can find through this link:
NCDOT Division of Bicycle and Pedestrian Transportation (click "maps" at the top). You can probably find a house not too far from some of these trails.
The Triangle MTB group has a lot of information on local mountain biking here:
TriangleMTB.com, your source for mountain biking in Raleigh-Durham-Chapel Hill
You can find group rides through local bike shops like Spin Cycle and Cycling Spoken Here in Cary. I know some weekend rides go out from the A.E.Finley YMCA in North Raleigh too.
Here's the link to "Get Out Get Fit" that Dire Wolf mentioned too: http://blogs.newsobserver.com/joemiller/ (broken link) . I just recently discovered it and it is a great resource.