The State Museum (right next to the children's museum)
The Columbia Museum of Art (downtown on Main)
The Statehouse
Columbia Marionette Theater
Trustus Theatre
Town Theater
Congaree National Park
Harbison State Forest
Finlay Park (downtown)
Riverfront Park (downtown)
Three Rivers Greenway (http://www.riveralliance.org/3rg.htm - broken link)
There might also be something going on at the
Koger Center or the
Colonial Center as well. And I would suggest taking a tour of the city's historic sites sponsored by the
Historic Columbia Foundation.
As far as some of the nicer subdivisions, those would be in the Lexington area (U.S. 1), the Irmo area (you can take Harbison Blvd. exit off I-26), and the Northeast area (Two Notch Road would get you there).