The information from mi127 is about an ICOCA Card, which is an IC transportation card.
An ICOCA (not "Icocca") Card can be used in Kansai Region on almost all railroad lines (including all Osaka Municipal Subway lines and all Kyoto City Subway lines).
In Kyoto City, IC transportation cards are not accepted on Kyoto City Bus lines, Kyoto Bus lines or Eizan Electric Railway lines.
An IC transportation card does not bring you a discount in particular, but it can save your time and trouble on purchasing and handling paper tickets for short-distance rides.
You can check "Basic Use" in this article.
http://www.japan-guide.com/e/e2359_003.htmlmy itinerary is below: When in Day 1 is your flight to KIX scheduled?
What would you visit in Kyoto City, in Nara City and in Osaka City?
When in Day 11 is your flight from KIX scheduled?
[Options of accommodations]
I wonder why you plan to stay first not in Kyoto City but in Osaka City even though you would visit Kyoto City first. (Maybe in order to see your Osaka hotel in preparation for later nights?)
I wonder also why you plan to head to Mount Koya not from Osaka City but from Kyoto City even though Osaka City is much closer. (Starting from Kyoto City, you travel via Osaka Prefecture or Nara Prefecture to Mount Koya in Wakayama Prefecture.)
If you can modify your hotel bookings, it sounds better
to travel first straight to Kyoto Station unless you might clear your immigration procedure too late for you to catch the last JR "Haruka" train
and also to make a roundtrip from Osaka City to Mount Koya.
(Even if you cannot find nice accommodation near subway Shinsaibashi or Nagahoribashi Station, if you find one in Osaka City near a station on subway Midosuji or Sakaisuji Line, you can make a Mount Koya trip in much shorter time than from the hotel in Kyoto City.)