Maybe by the end of my trip I will be familiar enough with Japan's transit systems that it won't be difficult?I think so :). Both of the bus and trains have their pros and cons. I would prefer trains in this case, since buses (Kyoto city bus 17 and Airport limousine) tend to be slower and less reliable due to possible traffic jams. Train transfers might at first site seem to be complex including 6 transfers, but actually you need four transfers, as will be explained.
1. From Kurama to Demachiyanagi, you have no choice. This should be only practical route. Kurama is the first station of the Eizan line while Demachiyanagi is the final station so you can't make mistake. If you have train IC card such as SUICA or ICOCA, you can use them from 16th March.
2. Demachiyanagi to Gion-shijo. Get out of the Eizan line gate using your IC card. Demachiyanagi station of Keihan line is on the second underground floor.
https://www.keihan.co.jp/en/station/pdf/180_demachiyanagi.pdfEnter from Eidenguchi ticket gate on the first underground floor using your IC card and then get down to the second underground floor. This is the first station of Keihan line, so you can't choose wrong train. From Demachiyanagi to Gion-Shijo, there are about 12 trains per hour, of which 6 are the limited express trains (every 10 minutes). These limited express trains are almost always (probably always, but I am not sure) the fastest, taking only 4 min. You don't need express fee on Keihan line. Get off at Gion-Shijo, just touching the card.
3. Gion-Shijo to Kawarawachi (Hankyu line) by walk
https://www.keihan.co.jp/en/station/pdf/162_gion-shijo.pdfGo through north ticket gates and get out of the station from Exit No. 4. Walk toward Shijo Ohashi bridge and you will see the Hankyu Kawaramachi station.
4. Kawaramachi to Minamiibaraki with a transfer at Ibarakishi
Find Kotocross Hankyu Kawaramachi
https://goo.gl/tbu6Seand then go down through Exit 3.
http://www.hankyu.co.jp/global/common/pdf/station_guide/kawaramachi.pd...Take limited express to Umeda (Osaka) and get off at Ibarakishi (You don't need to pay the express fee as well). Your connecting sub-express train is waiting on the same platform (opposite side). The transfer time is only one minute, but you don't need to worry, since you just need to walk about 10 meter.
5. Hankyu Minamiibaraki to Monorail Minamiibaraki
I can't find good map for this section, but you go up the stairs (or you can use escalator or elevator), go out through the ticket gate and then walk about 3 min to Monorail Minamiibaraki station.
https://www.youtube.com/watch?v=2kQM9OD3CeY6. Minamiibaraki to Osaka Airport by Osaka Monorail
https://www.youtube.com/watch?v=IKBuUg6_RMUFrom 12'00" you can see the view from the front window from Minamiibaraki to Osaka Airport. This is my favorite train in Kansai area. Note the traffic jam on the Highway (opposite direction though) :).