Ohio Statehouse, Pearl Market

Normally, one doesn't think of downtown Columbus as a hotbed for children's activities, but the Statehouse can be lots of fun. Built in 1839 and recently restored, the statehouse offers free tours, great architecture, and a bit of history. You may have to dodge school groups on field trips (or you could join them & eavesdrop on what they're learning), but you and the kids will learn tons of Ohio history, see some art (scale model of Columbus made from Legos was recently a big hit), and maybe see a real live astronaut pass through (John Glenn?). Plus the cool, pastoral building provides a break from the oncoming summer heat.
When you're done (on Tuesdays, anyway) head across the street to the Pearl Alley Market. There's food, home-made crafts & clothing, and live music, as well. Most vendors are local to Central Ohio, so it's a great way to support the local economy, and get some delicious soft-serve ice cream.

Ohio Statehouse: http://www.ohiochannel.org/your_state/ohio_statehouse/index.cfm
Pearl Alley Market: http://www.downtowncolumbus.com/pearlmarket/