Gwangju National Museum is the main place to get historical context for the wider Honam region. Its collection covers prehistoric material, Buddhist art, ceramics, and objects from the Baekje, Silla, Goryeo, and Joseon periods.
The museum is especially useful because Gwangju's modern story is so strong that older history can disappear behind it. Pair it with the nearby art institutions in Jungoe Park if you want a slower cultural half-day.