Kirk had recommended the Yanaba in Kosa Machi below, as do I - used to take my kids there when they were tots. If you do venture out there, here are three other nearby attractions that very much deserve a visit. First is the onsen Samata no Yu, which has both inside and outside baths. Be sure to take some time walking the grounds along the river to enjoy the view. http://samatanoyu-misato.jp/
Second is Futamatabashi, one of a collection of bridges a short walk downstream from the onsen. These bridges, which date respectively from the Edo, Meiji, Taisho, and Showa eras, demonstrate how Japanese engineering has progressed. In particular, the Futamatabashi is famous for the heart-shaped shadow it casts when the sun is in the right position in the morning. http://www.town.kumamoto-misato.lg.jp/q/aview/72/1056.html
Finally, there is the Shakain Misakayū Hodō, better known as 3,333 dan, or 3,333 steps - one of the longest stairways in Japan (or THE longest, if you take them at their word). If you have not climbed this wonder, you should. It starts from a narrow valley and ends at the Shakain Temple at the mountaintop; in fact, there are quite a few more steps than 3,333, but you’ll stop counting after a bit. The views are incredible, and autumn is the best season to visit, with changing leaves, persimmon trees, and pleasant temperatures. http://kumanago.jp/event/?mode=detail&id=430000000299
So - do the stairs, take an onsen, have some fish, and head home - a very busy but unforgettable day trip. Please let us know if you have any questions about these locations. - William