There're several places where you arrive in Fukuoka.
There's free shuttle bus service between International and Domestic terminals in Fukuoka Airport. Domestic terminal has a subway station. JR Hakata Station and Tenjin can be reached in 10-15 minutes.
Nishitetsu Fukuoka Tenjin Station is connected to Tenjin Underground Mall and onto subway Tenjin and Tenjin-Minami stations.
Hakata Port Ferry Terminal (terminal 2) is where you arrive from Iki Island, Tsushima Island, and Goto Islands. Note that ferry from terminal 1 goes to Uminonakamichi, Saitozaki, Shikanoshima, and Genkaishima Island, all within Fukuoka City. Nishitetsu Bus goes to Tenjin, Canal City, and JR Hakata station. Ride a bus to Bayside Place when you go to this Port Ferry Terminal.
Hakata Port International Terminal is a building next to Marine Messe. Nishitetsu Bus goes to Tenjin and JR Hakata station. Ferry and jet foil come from Busan. Taxi fare is 1,500-2,000 yen to JR Hakata station and Tenjin area.
Fukuoka has 2 bus terminals, one next to JR Hakata station and the other in Tenjin. Both have expressway bus connection to other cities in Kyushu, and some to Honshu and Shikoku.
Following is the rough estimates to other places in Kyushu.
Kitakyushu (1.5hr by expressway bus, 15 mins by Shinkansen, 1hr by expressway drive)
Karatsu (2hr by subway and JR Chikuhi Line, 1.5hr by expressway and regular road drive)
Nagasaki (2hr by JR Limited Express Kamome)
Kumamoto (1hr by Kyushu Shinkansen, 1.5hr by expressway drive)
Beppu (2hr by JR Limited Express Sonic, 1.5-2hr by expressway drive)
Yufuin (2hr by JR Limited Express Yufuin no Mori, 2hr by expressway bus, 1.5hr by expressway drive)
Kagoshima (2hr by JR Kyushu Shinkansen, 2.5-3hr by expressway drive)