Fukuoka has many nice places to see - those who don't know them keep repeating the same nonsense that there aren't. It's only 20 minutes by train from Hakata Stn to the Nanzoin Temple, which is one of the best you will ever see in Japan.
http://bit.ly/1KAm8cUhttp://www.thejapanfaq.com/fukpic.htmlThere are also several preserved old villas and traditional gardens, like Rakusuien, Yusentei, Shofuen, the Ohori Park Teien, and Nokonoshima Island Park. If you are there in early April, there are several very good places to see the cherry blossoms, which are not overrun by people like in bigger cities or Kyoto. One of the best yet completely off the tourist radar is in nearby Kasuya, at the Kayoicho Park.
https://youtu.be/pYKNmvevAC4If that is too early for you, I suggest you try the last part or end of April. That is azalea season, and there are some very good places to see them - Mifuneyama in Saga is wonderful, and so is the Daikozenji Temple in Kiyama. Kurume, about 30 minutes south of Fukuoka by train, is also famous for them, as well as the giant Naritasan Jibo Kannon Great Buddha.
https://youtu.be/G1SBtmqXteEYou also have a shot at seeing the wisteria - which at its peak is just heaven. The end of April and Golden Week (a busy holiday season from the end of April) can get busy, but the Kawachi Fujien in Kitakyushu is one of the best places to see them. Getting tickets and getting there can be troublesome though. There is an equally sized place without the crowds in Yame, at the Kurogi Ofuji.
https://youtu.be/23MpyQPPhAwHow far you want to go matters a lot though - or really, the time it takes. If you get a Kyushu Rail Pass, you could get over to Kagoshima in 90 minutes. Nagasaki is one of the most interesting cities in Japan, and best as an overnight.
http://bit.ly/2ob4u8iAlong the way is one of Japan's most beautiful yet unknown places, the Yutoku Inari Shrine. You may have seen Fushimi Inari in Kyoto, but this place doesn't have the tourist mobs and is in some ways even more enjoyable.
https://youtu.be/2UU_1Aip_l4Nearby is Unzen with some good hot springs, the Biopark was mentioned, plus at good flower seasons like then, Huis Ten Bosch is very beautiful.
https://youtu.be/JIKAjc5MfPsAnother great activity is the 99 Island cruise in Sasebo.
http://travel.sasebo99.com/en/99islands.phpThere is no shortage of great places in Kyushu.