The Ishin Furusato-kan, to give it its Japanese name, sits on the riverbank where many of the architects of modern Japan lived as boys. It is a small, friendly, well-translated museum and the right first stop for anyone trying to make sense of why this provincial Satsuma capital matters so much in Japanese history.
The basement holds a 25-minute animatronic theatre that walks through the events of the Restoration in good English subtitles; the upstairs galleries cover the lives of Saigo, Okubo, and the nineteen Satsuma students smuggled to Britain in 1865. Allow an hour. Combined tickets are sold with Sengan-en and with the Reimeikan if you plan to see all three.