Getting to Kinosaki onsen is actually pretty straight forward and it simply takes a train ride from either Kyoto station or Osaka station (which will be the case in the most cases).
I recommend getting on the Limited Express Kinosakionsen that will get you straight to Kinosaki onsen.

You can buy a ticket and reserve your seat through official JR website (you will need to choose KINOSAKI/KONOTORI/HAMAKAZE train with the Kinosakionsen as the arrival station)

📍 Kyoto station > Kinosaki Onsen station (google maps link)

• At the time of making this guide, the earliest train leaves at 11:25 am which allows to not rush at all in Kyoto and have a nice breakfast, check-out from your hotel / airbnb and have an early lunch. You will get to Kinosaki at around 2 pm which is perfect for this itinerary.

• If you would like to arrive a bit earlier then you can take the same route but look for the one where you make a transfer at Fukuchiyama station. Do not hesitate to ask the train station staff for help and directions during your transfer as there will be just 5-7 minutes before the 2nd train leaving.
  

📍 Osaka station > Kinosaki Onsen station (google maps link)