The Tallgrass Prairie National Preserve, about 90 minutes southwest near Strong City, is the most compelling day trip from Topeka. It protects one of the last large expanses of the native tallgrass ecosystem that once covered much of the Great Plains. The Spring Hill Ranch on the preserve is a well-preserved 1880s cattle ranch. Bison were reintroduced in 2009 and can be seen on guided hikes.
The Flint Hills Scenic Byway runs through the same region, following a stretch of rolling prairie that looks very much as it did before European settlement. This is one of the few places in the United States where you can get a sense of the pre-agricultural landscape of the interior.
Kansas City, 65 miles east, offers a full city complement of museums, restaurants, and jazz history if Topeka leaves you wanting more. The Nelson-Atkins Museum of Art in Kansas City is one of the best art museums in the Midwest and worth a half-day.