If you wish to only travel to outlying towns of the Kunisaki,

Usa (Home of the famous Usa Shrine) or Kitsuki, there are trains available.

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• From Beppu -  local, limited express train to Kitsuki or Usa, approximately 25 minutes and 460yen to Kitsuki, 45 minutes and 840yen to Usa.

• From Kokura – Hourly departures, stopping in Usa (100 minutes, 4300yen) and Kitsuki (110 minutes, 4900yen)

 

However, to venture further into the Peninsula and its rich History, it is impossible by train, you will have to switch to bus. This is most easily done in Oita or Beppu. If travelling from other parts of Japan, Kokura is a Shinkansen (Bullet Train) station and therefore easily accessible from all over Japan.

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• From Kokura to Oita – 1 ½ hours, 4,000yen, limited express sonic train.

• From Hakata to Oita – 2 ½ hours, 5,500yen, transfer at kokura for the sonic train.

 

Details: https://www.jrkyushu.co.jp/english/time_table/time_table.html