Nagasaki to Kumamoto via Unzen

So I am planning our second trip to Japan. We would like to travel from Nagasaki to Unzen, stopping on the way at Shimabara. Onwards to Unzen for a 1 night stay, and then travel to Kumamoto. Has anyone done this by public transport? Is it viable? Your advice is highly appreciated. I tried searching online but could not understand the timetables that came up as they were in Japanese.

It is quite easily doable if you do not venture off the main bus and train routes.
I personally would take trains to Shimabara, then the bus to Unzen and then the bus and ferry and another bus to Kumamoto. All these buses, trains and ferries are operating about hourly, so, it should be doable.

https://www.kumamotoferry.co.jp/lang/?lang=en

https://en.shimatetsu.co.jp/bus/
A few years ago, I travelled the opposite direction of yours: from Kumamoto City to Shimabara Gaiko (port) by ferry then took a train to Shimabara. From there I boarded on a bus to Unzen, after looking around the area for a few hours I moved to Obama Onsen by bus where I enjoyed staying at onsen ryokan and by the sea there was a ashiyu, where you could soak your feet in onsen water. I stayed there over night then the following day I took another bus to Nagasaki.
I liked the bus ride. You have to be careful about fare payment, if I remember I had to pay in cash and some new bill the machine did not accept. I am talking about the section between Shimabara to Unzen. The thing may have changed since then.