[IC cards]
which IC card can be used in all the cities on my itinerary?Yufuin Station seems to be outside the usage area.
https://www.jrkyushu.co.jp/railway/routemap/rootmap_num.pdfSuica cards and PASMO cards
are issued by Tokyo-based companies.
Due to shortage of chips,
currently the sales of Suica and PASMO cards
are basically suspended, practically limited.
ICOCA cards are issued by JR-West.
You can get one at a JR-West ticket counter
of stations such as Kansai-airport, Ōsaka, Shin-Ōsaka and Kyōto.
(In Shin-Ōsaka Station / Kyōto Station,
JR-Central manages the Shinkansen zones and counters;
there is a JR-West ticket counter
near East Gate / Central Gate.)
[Rainy season]
I'm considering to travel in March or April, when does the rainy season generally start? In May or June.
Usually it's not constantly rainy through the rainy season.
[Routes]
I've just searched the flights from Sydney, Australia to Japan and Tokyo seems to be the first stop. I'm looking for the shortest travel time and with the least transfers of transportation between the cities. There seem to be connections
five days a week
to Kansai Airport [KIX]
with one transfer at Cairns Airport [CNS] in your country.
Compared to going via Tokyo Haneda [HND],
this sounds more suitable for your purposes;
the first flight is much shorter,
and you, regardless of companies of your fights,
make a transfer without getting through the immigration control.
i- Fukuoka to Yufuin (stay overnight) In Fukuoka City you make railway transfers
at Hakata Station.
Please don't search for Fukuoka Station there....
- Yufuin to Kyoto via Fukuoka (stay 2 days) That requires about 5 to 5 and a half hours.
It sounds better to plan to stay somewhere in the middle.
Even the Shinkansen may be delayed, depending on weather conditions.
- Yufuin to Kyoto via Fukuoka (stay 2 days) / (via Kokura to Osaka seems to have multiple transfers..)The both routes have Kokura Station.
On the former route, at Kokura you are already on your Shinkansen train.
On the latter route, you make a transfer at Kokura to the Shinkansen.
: Yufuin -- Ōita -- Kokura -- Shin-Ōsaka.
Shinkansen trains run not via Ōsaka Station;
they use Shin-Ōsaka instead.