Train
Timetable and Route Search for all Japan, including ticket prices, you will find on JR West
About the total fare of your tickets, you always have the two items. Basic fare plus seat fee, a seat reservation costs extra. Off season and on weekdays it should be ok without seat reservation, there are always a few free 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 