Athens Memorial Park has a few very easy paved trails to walk on, a day use area for picnics, a free zoo, and a couple of miles of scenic singletrack trails. It's so hard to believe that some of these trails are right in the Athens city limits!
Located between Lumpkin St. and Milledge. See Google Map.
http://www.tripleblaze.com/trail.php?c=3&i=2258