Kyushu Shinkansen is running in Kyushu island, connecting Hakata station of Fukuoka city and the fourth largest city in Kyushu, Kagoshima via Kumamoto, which is the third largest city.
There are three types of train running. Mizuho which is fastest one, stopping only at Hakata, Kumamoto and Kagoshima-Chuo. Sakura which stops at more stations including Shin-Tosu, Kurume and Sendai. Tsubame is the local which stops at all stations. All Mizuho and most of the Sakura run through Sanyo Shinkansen sector to Shin-Osaka station, but Japan Rail Pass holder cannot take Mizuho trains as same as Nozomi trains on Tokaido/Sanyo Shinkansen.
In case you are going to Nagasaki from Osaka or Hiroshima, you should take Sakura train to Shin-Tosu rather than changing at Hakata station to limited express Kagome.