kyushu transport

Hi. We (2 adults) will travel in Kyushu via public transport in October. Will arrive & depart from Fukuoka. Would love to travel to Nagasaki, Kurokawa Onsen, and Yufuin/ Beppu. Will stay about 1-2 days each location.

What would be the most efficient order for commuting? Preferred to be budget. No JR pass.

Thanks!

I did a trip in '16 visiting those places. I'm not saying that it is the most efficient, but the order that we went was :

Fukuoka to Yufuin by train (nice trip, actually);
Yufuin to Kurokawa Onsen by bus;
Kuokawa Onsen to Nagasaki by bus/train (now we actually prefer train travel so I changed to train from the bus as soon as I could and I think that overcomplicated it. It would have been better to stop at Kumamoto at this point and go to Nagasaki from there but it was just after the earthquake, and we wanted to go to Huiss Ten Bosch for the Nagasaki Bio Park anyway);
Nagasaki to Fukuoka by train (actually we stopped for a few nights at Karatsu. The train trip from Karatsu to Fukuoka is really nice).

It's not the JR Pass, but there is a JR Northern Kyushu Area Pass that may suit your itinerary.
http://www.jrkyushu.co.jp/english/railpass/railpass.html

Kurokawa Onsen is not accessible by train, so bus is the only public transportation option.

Get the Kyushu northern bus pass SunQ.

I'd either do Fukuoka to Yufuin/Beppu, to Kurokawa Onsen, to Nagasaki, back to Fukuoka. Or in reverse.

Thanks everyone for the help.

This is another story.
Kurokawa Onsen is the rural atomosphere type, Yufuin is the suberb type
though I like the both, there are somewhere same types.

The following is the rustic type, if you like.
You can see the mountain and stars from the Rotenburo.
https://www.tripadvisor.jp/Hotel_Review-g1121560-d1367843-Reviews-Kuju...