The previous post contains some great suggestions. I agree with the BBQ comment. There are lots of places to go. Everyone in KC has a favorite, but they are all good.
If you are interested in art, particularly modern and contemporary art, check out three places. The Nelson-Atkins Museum of Art is one of the top art museums in the U.S. It's a 10 minute walk east of Country Club Plaza and a 10 minute bus ride south from Union Station, and easily worth a morning or afternoon of your time. The Kemper Museum of Contemporary Art is worth a visit and is two blocks northwest of the Nelson-Atkins Museum. Finally, the Thomas Hart Benton Home And Studio State Historic Site is halfway between Union Station and the two museums in the Roanoke section of Kansas City. It contains some of Benton's works, his studio and is another good way to spend a couple hours.
Continuing the art by Amtrak theme, a hour and and a half ride east on the Missouri River Runners will take you Sedalia, MO and the Daum Museum of Contemporary Art. This is one of the best contemporary art collections in the country (really, honest, straight up, for sure). It's on the campus of State Fair Community College, about a 10 minute ride from the Amtrak Station. There is not much else to see in Sedalia, so go during the Missouri State Fair in August, spend some time at the Fair, enjoy a couple hours in the air conditioning at the museum, eat dinner, and take the train back to Kansas City. Jefferson City is an additional hour east of Sedalia. The Jefferson City Amtrak station is part of the Jefferson Landing State Historic Site and located in the same building as the Elizabeth Rozier Gallery which has rotating exhibits related to Missouri history. The state capitol building is a few blocks away and the tours of the capitol include the Missouri State Museum and cover history of a statues, murals, and paintings that were commissioned for the building. The high point of the tours to me, is to the House Lounge and a chance to see Thomas Hart Benton's "A Social History of the State of Missouri." Lots of restaurants are nearby, and it's easy to spend an afternoon just at the Jefferson Landing Site and the Capitol.