First, check on group rides leaving from Cycle Center, which is right across Broad Street from the East Campus, next to Whole Foods:
http://www.durhamcyclecenter.com/grouprides.shtml
You already have the Tarwheels link. Most Tarwheel rides are B/C pace, which is too slow for you, but there is generally an A ride out of Wilson Park in Carrboro on Saturdays.
You should also check in with Clean Machine in Carrboro, which is the focal center for Doughnut Boys Racing, the most active Durham/Orange County group of competitors. They operate a weekday time trial every week. While in Carrboro, stop into Performance Bicycle (remember, this is the home of Performance); a number of group rides leave from there.
Now, as to solo routes, I’d recommend heading west and then either north or south. From East Campus take Campus Drive past the West Campus to Cameron Blvd./751, where you take a right. Once under 15/501, go left on Erwin Rd.
For southerly destinations, continue south and turn right on Mt Sinai. After a few hills, you’ll T into NC 86, where you go right. Next, turn left on New Hope Church Rd., which, once over I-40, is the feeder to a lot of great Orange County cycling. If you crave hills, for example, be sure to turn right on Borland Road. For your longer rides, you could go to Saxapahaw or Snow Camp.
For northerly directions, go right on Cornwallis from Erwin and follow until it crosses US 70 and becomes Pleasant Grove. Much fine riding north most of the way to Virginia between Hillsborough and Roxboro.
David