Shimonoseki on Honshu is a short train ride from Kokura. Additionally, StarFlyer has a promotion where, if arriving from or going to Tokyo on any of their flights, they will offer taxi service between Kitakyushu Airport and Shimonoseki for only 1000 yen.
Hakata station in Fukuoka is twenty minutes by Shinkansen from Kokura, or about one hour by express.
