With that much time, you really should be able to cover Kyushu comprehensively. One could actually cover all of Jpn in a month.

I probably have different interest than you, but even then I would reallocate some of your dates to Fukuoka. I'd try to minimize having to check in/out of hotels. There's so much to do/eat in Hakata and you could do day trips via train from Hakata to Yanagawa, Dazaifu, and some of the places in Saga such as Karatsu and even Arita. 3 days in Shimabara is a bit of an overkill. In line with what jib said, cover the first part of the trip (Hakata and day trips from there) via train, staying the first 5-6 nights in Fukuoka and then heading on down to Nagasaki to spend a few nights there. From Nagasaki, you can do one or two train day trips including Sasebo with Kujuku-shima cruise. I would then pick up a rental car in either Nagasaki or better yet Isahaya, then head on down to Unzen to check out the volcanic steam and stay at an onsen place. Then drive down the next day to Shimabara. The town is very nice with water flowing everywhere and you should also check out the castle there, but I really don't think the town is worth more than half day (we saw what we needed to see in <3hrs). You can then take the ferry to Amakusa and drive around and make your way towards Kumamoto and onto Aso and Kurokawa Onsen. Then drop off your car in Kagoshima. I've not been to Kagoshima or Miyazaki, so I can't help you there. I think it's okay to give Oita a miss as your schedule has it (Yufuin would be the main attraction).