Do you want paved, gravel, or singletrack?
For paved, there is a great network of greenways. I did not click on the link above, but it seems to be covered there.
For gravel, if you want flat, try the American Tobacco Trail (ATT). Rails to trails with lots of multi-use going on there. It is very flat, which might be a good start. Umstead is great too, but much more hilly.
If you want singletrack (actual trails) then I would start with Lake Crabtree County Park, or Harris Lake County Park. Both have beginner to intermediate trails. Both have about 8 miles approximately, and that should be enough to keep you busy for a while. If not, turn around and do them in the other direction.
A great biking resource for our area is
www.trianglemtb.com (you will see me over there too). They are mostly focused on singletrack, but you can get info for any type of riding there.
Biking is a huge passion of mine, if you have any questions, please don't hesitate to ask away, I do a LOT of riding in this area, and around the state.
Have fun!!!!!!!
EDIT>>>> They have a very active forum over there on trianglemtb. They also have a section on trail descriptons and directions on how to get there.