It looks like you are leaving in about a week and you still have no definite plans? You need to get set as soon as you can - and hotels might be booked up as well at the cheaper and more conveniently located places. Also, you didn't say where you are arriving from, but if you are taking a long international flight, your first day you might not feel like doing much. You are traveling during the cherry blossom season, so there will be some places with a lot of people.
You don't give any info on your interests, but aside from the cherry blossoms, one of the best sights in the region is the Nanzoin Temple, about 20 minutes by train from Hakata Station. http://www.thejapanfaq.com/fukpic.htmlhttps://goo.gl/X3LDaE
On the way back, you could also stop off at Kayoicho Park, which is beautiful during the cherry blossom season as well as being virtually unknown to everybody. For Kumamoto, you didn't specify if the city, or the region - like Aso. The city is just 40 minutes away by bullet train, but the castle is still closed after the huge earthquake, and one of the few other good sights is the Suizenji Garden - nice, but maybe not worth the trip there just for that. Aso is possible but very time consuming to get to. And the crater may be closed when you want to go. You can look at this sitehttp://www.aso.ne.jp/~volcano/info/
as some indication for the day (though it could change en route, and usually not for the better). There is one other option if you don't mind paying more - you could get the full Kyushu Rail Pass and take a day trip down to Kagoshima. It's 90 minutes one way by bullet train, and you can walk by the live Sakurajima volcano, not to mention the nice Senganen Garden.https://goo.gl/ir1cA5https://www.japan-guide.com/e/e4600.html
Nagasaki is definitely worth it - it's really the most interesting city on the island. The food is also excellent.https://goo.gl/hZVjBi
Along the way, you could also see the beautiful Yutoku Inari Shrine.https://goo.gl/dnqT25
You might visit Nagasaki on Day 5, and see Dazaifu on Day 4 with Fukuoka (and perhaps Yanagawa also if you like).
As for hotels, you don't how many, your budget.... you could always try a business hotel chain like Toyoko Inn. Nothing fancy, but clean, quiet, conveniently located, good internet and free breakfast buffet.