Trans-Kyushu Limited Express (Kyushu Odan Tokkyu=九州横断特急) connects the most famous hot springs resort, Beppu (別府)/Oita (大分) with Hitoyoshi (人吉) via Kumamoto (熊本). The biggest volcano, Mt. Aso is located between Oita and Kumamoto. This is the train that you need to take to Mt. Aso. Kumagawa (くまがわ) is operated between Kumamoto and Hitoyoshi on same route as Trans-Kyushu Limited Express.
Kyudai line was damaged by rain storm in the last year. But it has been repaired on July 14th. Yufuin no Mori is now operated by normal timetable.
These trains connect Hakata (博多) with Yufuin (由布院) and Oita (大分) or Beppu (別府). Yufuin is one of the most popular hot springs resort in Kyushu. This is the train that you need to take when you go to Yufuin. Of course these train’s name are named from Yufuin. “Yufuin no Mori” means Yufuin’s forest. These two trains are operated on same route and can be used by Japan Rail Pass. But Yufuin no Mori is operated by the specialized car, KIHA71 or 72 series. This express train is one of the best in JR Kyushu train. So I strongly recommend to take Yufuin no Mori, not Yufu.
This limited express train is one of most popular limited express train that is serviced by JR Kyushu. This connects Hakata and Kokura with Oita and Saeki. If you go to the popular hot springs resort, Beppu, this train is that you need to take after you get off Shinkansen at Kokura or Hakata. Of course this train is fully covered by Japan Rail Pass.