The area has a number of good places. It's a little unclear how far you are willing to travel though. If you don't find sights in northern Kyushu too far, a Northern Kyushu Rail Pass might save you a lot and help you get around.https://www.japan-guide.com/e/e2361_05.html
Fukuoka itself has a number of good places to see - though people who don't know them recycle answers that there is not much to enjoy. One of the best temples in Japan is just 20 minutes by train outside the city and free. The Nanzoin Temple would be unforgettable for you. Plus there are others as well, including Japan's biggest wooden carved Great Buddha at Tochoji (also free), some very nice traditional gardens at Yusentei, Rakusuien, and Ohori Park, plus some killer night views from the Fukuoka Tower and Minami Park Tenbodai.http://www.thejapanfaq.com/fukpic.htmlhttp://bit.ly/1KAm8cU
There is also a great 2 day subway pass you can use
and many places are close to one of the subway stations.
One place you should not miss for your trip is Nagasaki - it is the jewel of the island and well worth some of your time. There really is no other place with its history and mixing of cultures from centuries ago. Be sure to try the chanpon or saraudon as well for lunch or dinner. Some great places are the Glover Garden, Dejima, Koshibyo Shrine, Peace Park/Museum, Gunkanjima, and in the evening, the view from atop Mt. Inasa for the "$10 Million Dollar View".http://bit.ly/2ob4u8i
Kumamoto took a big hit with the earthquake in 2016 and one of its main draws, its castle, is still closed. The Suizenji Garden there is still very nice to see, but your time might be better used elsewhere. Mt Aso is another possibility, especially with a rental car.http://bit.ly/2HopvoC
Kurokawa was mentioned which is one of the best outdoor hot springs in the country. It is rather time consuming to get to by bus (no trains). There is also Beppu, which also has some other sights, and Yufuin, although again due to repairs is harder to reach. One much closer to Fukuoka is just 15 minutes away and where you change trains to get to Dazaifu - there is Futsukaichi, and while it is not nearly as good, it does have a history going back over 1300 years.
There are some other good outings the other direction as well - Kitakyushu is more of an industrial city, but there is Kokura Castle, the Mojiko Retro area, and one of Japan's best city views is from atop Mt Sarakura. Just across the straight is Shimonoseki, which has some very nice sights as well, and is given short shrift by all the guides.