I think Kyushu is a great place for first time visitors to Japan. There is no reason Tokyo and Kyoto have to be places to visit for first time visitors to Japan.

Shinkansen (Bullet Train) now runs all the way to Kagoshima (Kagoshima-Chuo station, 鹿児島中央) where you can go from Hakata (JR station at Fukuoka is called Hakata, 博多) to Kagoshima-Chuo in one hour and twenty minutes, from north Kyushu to south Kyushu in less than an hour and half. It will be easy to get around Kyushu by train.

Also, as a train geek I must say JR Kyushu has a lot of interesting trains running around. The best of all is Seven Stars in Kyushu (ななつ星in九州) where least expensive journey is JPY 180000 (about 1269 Euro) per person. I understand this may be out of your budget, but on the website look under Course description and will show train route on a map of Kyushu. These locations where Seven Stars in Kyushu train stops are definitely places to visit in Kyushu. Using Rail Pass you can visit those locations by JR easily.

If you have two weeks stay then you can visit outside of Kyushu easily. By Shinkansen (Bullet Train) Hiroshima is only an hour plus from Hakata (JR station at Fukuoka), and Osaka is two hours and half from Hakata. Both are easy reach from Fukuoka by Shinkansen (Bullet Train).

If you would like to get another stamp on your passport then there is a jet ferry service from Fukuoka to Busan, South Korea which is three hours journey. Also StarFlyer Airline flies from Kita-Kyushu to Busan, 50 minute flight. There is 9 a.m. departure from Kita-Kyushu and 8:05 p.m. departure from Busan. StarFlyer provide discounts fares and there are a lot of people take a day trip to Busan taking advantage of discount fare.

To get to Kita-Kyushu airport, from Hakata to Kokura 17 minutes by Shinkansen (Bullet Train) and 40 minutes bus ride from Kokura station to Kita-Kyushu airport.

Hope you enjoy the trip to Kyushu.