The Columns on the Quad of the University of Missouri (http://maa.missouri.edu/) campus are the second-most photographed landmark in Missouri, right behind the St. Louis Arch.
While on the Quad, visit the the Museum of Art and Archeology, (http://maa.missouri.edu/) which has a fascinating collection of everything between ancient Southeast Asian sculpture to paintings by famous Missourians.
Also on campus, close to the Quad, is the Ellis Library, which houses the State Historical Society of Missouri. (http://www.umsystem.edu/s...)
The interesting and unique architecture of the Columbia Public Library, located just west of downtown at the corner of Broadway and Garth Avenue, is a sight to see. While you're there, enjoy a good book and a cup of coffee from the coffee bar located in the lobby. (http://www.dbrl.org/colum...)
The award winning Shelter Insurance Garden is located off of Ash Street, just before Stadium Boulevard. The gardens feature many different types of plants and flowers, along with beautiful sculptures, ponds, and a historic schoolhouse circa 1800s.