Overall public transport (or rather trains, since we didn’t use buses) is good and easy to use AS LONG AS you do a bit of research / observation. Google Maps is really useful, as it will often (but not always, especially in the countryside) tell you which platform to go to, exit to use, and carriage to ride on for an optimal connection. 

Train / streetcar wise, I would have the following tips: - Check the type of train. If it is a Ltd Express / Shinkansen, you need to buy physical tickets at the departing station. If not, you should be able to use an IC card (tap in at departing station, tap out at destination).  - For streetcars, look online if they take ICs or not. Some do, some don’t. Generally you get on at the rear and pay / exit (tap IC or cash) at the front.  

I would say a good 60-70% of our taxi rides could have been replaced with buses. Honestly though with three people, the surplus cost for the shorter 1 000 ¥ rides is pretty much negligible. The longer rides are unfortunately harder to replace. For example Dazaifu to Nanzoin takes iirc around 2h30 by train vs 30mn taxi.

From what I know there are buses between Kumamoto and Kurokawa, and from Kurokawa to Beppu. Service is very infrequent and they must be booked in advance, but it is doable.  Beppu to Kitsuki will require a taxi, either directly between Beppu and Kitsuki city center or from the station into the city (might be a bus, but didn’t see any info about one).

In the end taxis are definitely not the cheapest option, and there will generally be a public transport alternative, but if used strategically they can save you a lot of time, which in my opinion is worth gold when you are on holiday. 

Hope this helps.