The Eno River is not very far (under 30 minutes from Raleigh/Durham border) but has great trails, none very long or steep with historic mill sites. The "River" is not exactly the Colorado and doesn't appear especially deep but it does look clear. It goes into the Neuse and then Falls Lake.
Here is where it is located: Map and directions
And some general information about the park:
Here is a trail map. I usually go off to the right on the Pea Creek Trail which then goes into other trails but I am just as happy to go in the other direction on the Cole Mill Trail.
As per Leslie's very helpful detailed instructions, let's meet at the entrance at the intersection of Umstead & Cole Mill (NOT the one at the very end of Cole Mill Rd) which would be the Cole Mill Access Area and not the Few's Ford Access Area. Cell phone is[masked]-5861. I hope I can get reception.
Meeting area trails
And, if you really like the area, check out the annual Festival for the Eno on the July 4th weekend. It's $15 a day.