By ship
If you're coming from Nagasaki, the numerous ferry services may be preferable. One such route invloves travelling via Shimabara. Take the JR line from Nagasaki city through to Isahaya then switch to the private Shimabara line. Its not JR but don't go through the ticket gate to get there, its tucked away on Platform 0 yes, in Kyushu they count from zero, go figure and you buy your tickets from the rather inquisitive drivers on the train. Two ferries depart towards Kumamoto from different ports along this line but the most convenient is likely the one from Shimabara Port itself direct to Kumamoto Port. Walking maps from the stations to the ports are scrawled at each train station. After a rather scenic 30-60 minute cruise, you'll have to connect to a bus to actually reach central Kumamoto city.
Thats 4 modes of transport total and the times don't always match up so allow perhaps 4 hours for the journey unless you carefully plan it out using one of the big transport books at JR Nagasaki's ticket counter. Its all in there somewhere, I promise. Expect the entire trip to only set you back roughly 2500 yen, most of which goes to the Shimabara train company despite being the shortest leg. English-language signage along this route is minimal. Incidently, the final Kumamoto-bound ferries depart at 8PM daily, and one daily train reaches Shimabara a few minutes after that so don't get caught out.
By plane
Kumamoto has an airport, recently renamed the Aso-Kumamoto Airport, from which shuttle buses make a one-hour jaunt into the town ¥670. There are flights to and from Tokyo Haneda, Tokyo Narita, Osaka Itami, Kansai International, Chubu International, Matsuyama, Naha, Amakusa, Shizuoka, and Seoul.
If arriving at Fukuoka Airport, there is a highway express bus service named Hinokuni-go that will bring you to Kumamoto for ¥2000 and takes roughly 2 hours. Alternatively you can take the subway to Hakata Station and take the JR Kagoshima Line see below.