From Osaka, I highly recommend to take the direct bus instead of trains!!
There is no direct train from Osaka!
Approx. 2hours 30mins, if you take at least one express train.
Fare: around 4510yen-5140yen(single adult),depends on trains
Here you can find all details of train time schedule and so on.
http://www.hyperdia.com/en/
Type in "Osaka" and "Amanohashidate" or "Miyazu".
Then you can add "search details"→"order"and you can choose what is your priority "time", "transfer" or "money".
From 1st April 2014, if you are foreign visitor, you can purchase Kyoto Sea Area Pass. The pass benefit a lot for most of you. Please check on my blog page for further detail of the pass.
KYOTO SEA AREA PASS.