Kyushu December itinerary

Hey guys!
After traveling in Honshu for over a month, I decided I should give Kyushu some attention as well.
In a few days I'll arrive to Hiroshima for a three days visit, and from there I thought to continue to Kyushu.
Sadly I have almost no knowledge of it, and I would love some tips on what I should see, and especially if it's even smart on my side to go there in December.
Rough itinerary by looking at the map:
Hiroshima - Fukuoka - Nagasaki - Kumamoto - Kagoshima - Yakushima - Kagoshima (obviously) - Miyazaki - Oita - Beppu - Kitakyushu - Hiroshima.
I know there are some mountains and reserves that I should try and get to (like Mt. Aso), but I'd love any recommendation and where should I get to it from.
*How is Yakushima in December, btw?
*So far in Honshu I've traveled by local trains and day/night buses. Is it going to work well (and cheaply) for me in Kyushu as well?
Thank you!

Here is a list of the best 30 in Kyushu in December by a company Jalan.
Huis Ten Bosch, Sasebo, Nagasaki
Mt Inasa, Nagasaki
Hakata station, Fukuoka
Kurokawa Onsen, Aso
JR Hakata City, Fukuoka
Hashima/Gunkanjima, Nagasaki
Dazaifu Tenmangu, Fukuoka
Udo Shrine, Nichinan, Miyazaki
Takachiho Gorge, Miyazaki
Kumamoto Castle
Suginoi Hotel, Beppu, Oita
Kirishima Shrine, Kagoshima
Mt Sakurajima, Kagoshima
Chiran Peace Museum, Kagoshima
Yufuin Onsen, Oita
Usa Shrine, Oita
Ohori Park, Fukuoka
Marine World Umi no Nakamichi, Fukuoka
Tengin Underground Shopping Area, Fukuoka
Marinoa City, Fukuoka
Fukuoka Zoo and Botanical Garden
Canal City Hakata, Fukuoka
Kannawa Onsen, Beppu, Oita
Sand Bath, Kagoshima
Umitamago Aquarium, Oita
Umi Jigoku, Beppu, Oita
Nagasaki Trams
Hirayama Onsen, Kumamoto
Beppu Onsen, Oita
Jigoku, Beppu, Oita
https://www.jalan.net/kankou/pro_010/?screenId=OUW1021

By their bar chart for Yakushima Island, I could read the most visited months are August and January. The least are December and February. In Japan, people could get a long vacation only in August (Bon) and January (Shogatsu). So Yakushima should be OK even in winter.
https://www.jalan.net/kankou/spt_46503ab2050007218/

Kyoshu is a highly populated island. You do not have to worry about transportation so much. It seems SunQ Pass would be popular.
http://www.sunqpass.jp/english/index.shtml