Hey - I don't have much to add, since the people above have given you decent info. Try http://www.trianglecycling.com
They do lots of organized rides when it gets warmer, and they also usually put up info about rides from various stores in the area. As for running, Tobacco Trail and Umstead Park are both great. I actually live over in Chapel Hill (I'm a grad student), but I have never heard of anyone having problems with safety in the area. Outside of just being smart about knowing where you are, etc., you should be fine.
If you ever want to catch up and ride/run/whatever sometime, let me know. When I manage to get out of the library, I always appreciate company! Hope you're enjoying lovely NC :)