suggestions for 12 days in Kyushu

Plan to arrive in Fukuoka and stay 1 night at the beginning and 2 nights at the end. Would like to spend 1 night in Nagasaki, 2 nights in Kagoshima, 2 nights in Miyazaki, 1 night in Takachiho, and 1 night in Yufuin. Would like suggestions for the route. thanks

I recommend the following route:
Fukuoka - Nagasaki - Kagoshima - Miyazaki - Takachiho - Yufuin - Fukuoka

I recommend to rent a car for the section from around Takachiho to around Yufuin and drive via Aso. It can also be made by public transportation, but it is complicated.

If you plan to use public transportation to get to Takachiho, get on the Limited Express Nichirin to Nobeoka, which is a little over an hour away. You will then have to board the local bus to Takachiho, which will also take a little more than an hour.

You should be able to purchase a round trip bus ticket to get back to Nobeoka. When you return, catch the next Limited Express Nichirin to Oita, and switch trains to Yufuin.