Hey, this is going to be mostly from the point of view of a tourist who is into history, but I hope it helps.
Check out the UNC Morehead Planetarium evening shows, especially if you have a romantic interest to go with you. I went there on what I consider to be my first real date. Great place to make some memories.
Also, the Carrboro Farmer's Market is a nice way to spend a Saturday morning. Lots of local produce and crafts.
The UNC Campus is quite pretty in itself if you want to take some rollerblades. The sidewalks are paved with red bricks. You just have to dodge all the squirrels. Some of the buildings date back to 1793.
If you want a typical (but unhealthy) North Carolina lunch, check out Sutton's Drug Store on Franklin Street. Before I became vegetarian, I used to eat there all the time. Nice people, old-fashioned menu. You won't leave hungry.
If you want something a little less atherosclerotic, you might want to check out the restaurants just as you enter Carrboro from Chapel Hill which tend to be more organic.
I love this city.
If you only plan to spend a day or so there and if you like history, I would go ahead and cough up the money to take the trolley tour of the city which leaves from the Savannah History Museum on MLK.
Also, the tour at Fort Pulaski on the way to Tybee Island was well worth it if you are interested in Civil War history.
As a tourist, I enjoyed Uncle Bubba's restaurant (Paula Deen's brother's restaurant). It's definitely touristy, but the food was good.
I wanted to have dinner at the Pink House, but it was booked.
For breakfast, I went to Clary's Restaurant on Abercorn. Typical southern breakfast fare. It was much more of a local hangout.
River Street is always a nice morning or early evening walk. Hot in the afternoon though.
Hey, I hope this helps and I wish you a safe and enjoyable road trip.