Kyushu route can consider below,
Day1 Fukuoka via SQ direct flight and reach in the morning -> Nagasaki (stay)
- Activate All Kyushu Rail Pass 18,000 yen
- Stay
http://www.jrk-hotels.co.jp/en/Nagasaki/#roomWrap
- Visit Glover Garden, Oura Catholic Church, Nagasaki Chinatown and evening / night go Mount Inasayama for sunset and night city view.
Day2 Nagasaki -> Fukuoka (stay)
- Morning visit Gunkanjima, follow by Nagasaki Peace Park and Atomic Bomb Museum.
- Evening take train go back Fukuoka.
Stay
http://www.jrk-hotels.co.jp/en/Hakatachuo/
Day3 Kagoshima -> Fukuoka (stay)
- Visit Sakurajima, Senganen Garden.
Day4 Fukuoka, take enough clothing / things enough for 1 night, the rest request to put in hotel as you will be returnning -> Beppu (stay)
- Visit Hells (Jigoku), then lepak at onsen resorts.
Day5 Yufuin -> Fukuoka (stay)
- Take train from Beppu and stopby Yufuin. Can visit Lake Kinrinko and shopping area along the main walking route.
- Can hoot cake from here if you're early, it get sold out fast
https://tabelog.com/en/oita/A4402/A440201/44000122/
- Can have lunch beside Lake Kinrinko
https://tabelog.com/en/oita/A4402/A440201/44000042/
Take Yufuin no Mori 6 back to Hakata
http://www.jrkyushu.co.jp/english/train/yufuin_no_mori.jsp
Day6 Fukuoka (stay)
- Visit Dazaifu area, evening go Momochihama Beach for sunset and
Fukuoka Tower for night city view.
Day7 Takachiho via highway bus -> Fukuoka (stay)
- Visit Takachiho Shrine, Takachiho Gorge.
Day8 Fukuoka (stay)
- Morning visit Uminonakamichi Seaside Park for some cycling, quiet place to cycle around, afternoon go Canal City Hakata for shopping.
Day9 Fukuoka, morning flight back -> SIN, reach afternoon
Been to 22/47 prefectures, Kyushu the nicest region that I've really enjoyed.