A good way to get from the railway station into the center is to take a tram. Also if you choose to stay in one of the 2 youth hostels, you will have to use it.
Of course, you can walk into the city center but it's a fairly long walk.
The Shinkansen is by far the most convenient way to travel around Japan. Unfortunately it isn't very cheap! Luckily there is a Japan JR Rail Pass available to foreign travellers. See the JR East website for details:
It must be bought abroad and stamped in Japan before you can use it. There are several options including one week (unlimited JR travel except for the super fast Nozomi service), two week, JR East and JR West passes. Check with your local travel agent for the best option or click on the link to see for yourself!
The Shinkansen only goes as far as Fukuoka city in Kyushu so you will have to change here to the Limited Express train (also run by JR).
Here is a link to a very cool page that lets you see the routing between cities/stations/ports within Japan. It will tell you the route and includes the cost and time. http://www.jorudan.co.jp/english/norikae/.
Kumamoto is on the JR Kagoshima line with regular trains passing through from Fukuoka (north) and Kagoshima(south). There are also trains from Miyasaki to the east. The Ariake Train is the service between Hakata (Fukuoka City) and Kumamoto and the Tsubame train runs from Hakata to Kagoshima with a stop in Kumamoto.
If you plan to do a lot of travelling by train in Kyushu I recommend that you buy the Nice Going Card for 500 Yen. This card allows travel over three days (weekend included) at a discount. Ask about the card and individual fares at any Green Counter at the JR Stations.
JR Kyushu also market a Kyushu Rail Pass which can only be bought overseas. The cost is 15,000 Yen for a 5 day pass and 20,000 Yen for a 7 day pass. It is only valid for local and limited express trains in Kyushu and not JR buses or the Shinkansen that runs from Fukuoka.
JAPAN SHINKANSEN MAP: Click Here
SHINKANSEN ROUTE MAP: Click Here
We were on a coach tour with All Nippon Airways (ANA). I forgot how we got here :P but it's definitely by coach :)))
I arrived by train to Kumamoto from Hakata. Kumamoto train station has all the usual facilities including a tourist office. I also used a café there on the upper floor.