Being a biker doesn't mean you have to clip in and pedal hard all the time. Area buses generally come equipped with "transit racks," designed to hold two bikes just in front of the driver's enormous windshield. It's perhaps the easiest way to get anywhere on a bike, so don't be intimidated by the seemingly unwieldy (and sometimes very rusty) contraption. A caveat: Each bus can only hold two bikes, so show up early and don't be shy.
When your bus stops, walk to the front. There's a latch just beneath the top bar of the rack. Squeeze it, and slowly ease it parallel to the ground. (Only let it fall if you like being yelled at by the person carting you around.) Pick up your bike, and place the front tire in the hole marked "Front Wheel Here." Put the other wheel in the other hole. Now look for a J-shaped hook. Lift it up and outward, and a rod will extend from the base. Put the rod on your front tire, near the fork. Make sure your lights are off, board the bus and enjoy the free wireless.
The whole process should take about 25 seconds, which is a lot faster than biking from Raleigh to Chapel Hill.