Bus
Information and reservation you will find here.
Train
Timetable and Route Search for all Japan, including ticket prices, you will find on JR West
To search for stations, it is better to use capital letters, some will not work otherwise. About the total fare of your tickets, you always have the two items. Basic Fare plus Seat Fee. Off season and on weekdays it should be ok with the unreserved seat. There are always a few cars. Searching for Fukuoka use the old name Hakata.
Flights
ANA domestic flights you will find here.
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JAL domestic flights you will find here.
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My Kyushu 