Uh, hm. Well, there's the National Park/historic site in downtown, with Andrew Johnson's tailor shop, homestead, etc. There are a lot of historical sites around downtown and there's a whole walking tour set up. I'm sure you could find more info about all that at the National Historic Site's welcome center.
One note to mouse's post, the Capitol Theater had to shut down due to lack of finances. NPAC is still going strong.
Welcome to the Nathanael Greene Museum
TN State Parks: Davy Crockett Birthplace State Park
http://www.nps.gov/anjo (A Johnson Historic Site; for some reason this link isn't working at the moment)
Historic Sites/Attractions
Walnut Ridge Llamas
If you're into nature at all I highly recommend utilizing various parks in Cherokee National Forest around Greene County as they are the "real deal." I would recommend Horse Creek Park or Paint Creek Park. If you're really interested lmk and I can tell you how to get there, as they are a bit out of the way for the average tourist.
Be warned that you'll be more satisfied staying in Greeneville if you're looking for a very quiet getaway without a lot of tourist attraction. If you're more of the "beach/Gatlinburg" crowd then you might do better looking toward Jonesborough/Johnson City. Jonesborough has much of the same feel/look as Greeneville but has been exponentially more restored and developed as a tourist stop for people coming through the Tri-Cities. While Greeneville is almost pure native, many people in downtown Jonesborough are from all over the state and even a few from outside it.
