There are lots of cyclists in the Triangle area (Raleigh-Durham-Chapel Hill). Bike commuting is not as widespread, although much more common in Chapel Hill. Traffic is heavy throughout the Triangle, but commuting is very doable if you are careful about choosing your routes and take adequate safety precautions (eg, bright clothes, lights). I commute year-round in Raleigh, which is more heavily populated than Chapel Hill and less bike friendly. There are very few bike lanes/paths in the area, but Chapel Hill has more than the other cities in the area.

For rentals, I would suggest trying the Bike Chain on Franklin St. Lots of bike shops in the area that you could find through google if the Bike Chain doesn't rent bikes. There is also a Performance Bike shop in Carrboro that might rent bikes.

If you rent bikes, I would suggest joining a group ride with one of the local bike clubs if possible. That would give you a better feel for cycling options and routes than just picking up and riding in a strange city where you don't know the roads. Here are links to local clubs:

www.tarwheels.org
www.ncbikeclub.org