OK, I think I'm going to recommend that you do not get a JR 7 or 14 day pass, but I need to know what airport you will leave Japan from.
First off, the 7 day JR pass works out to just over 4000 yen/day. so if there is a day where you are going to spend less that that, I'm not sure it is worth getting for that day. You first day is probably going to be getting from the Kansai Airport (KIX), to your hotel room in Osaka, and that may be it. If you stay in somewhere with interesting attractions nearby, you will just have to pay for the Airport transfer that day. There are several transport options to the Namba district for about 1000 yen.
http://www.japan-guide.com/e/e2033.htmlThere are several bundled passes for visiting some of the tourist attractions like Nikko, Hakone, and Mount Koya. I would get these, even if you plan to get a Jr Pass.
Many attractions, like the ones I have listed, and some cities, such as Kyoto and Hiroshima, will have privately owned operators that you will have to use. The bus service in Kyoto is better to get you many attractions around the city, and the trolley system in Hiroshima will get you back and forth from Peace Park. Since Hiroshima is a day trip, and if you plan to visit Miyajima Island, you may only need the Hiroshima trolley for one trip, Peace Park to Hiroshima Station. You may want to get a day pass for the bus system in Kyoto, but your might just pay for the one trip on the trolley in Hiroshima.
Anyway... Mount Koya train and cablecar are run by the Nankai private line, get the Koyasan Heritage Pass. Nikko is served by the Tobu private line, and once again, get a Heritage Pass. Hakone is several operators operating the five different mode of transport around the "Round Coarse" (train, bus, cable car, ropeway, and boat). Get the Hakone pass that covers then all. The Nikko and the Hakone passes have different "flavors", so choose the one that is most appropriate to you. Here are some links:
http://www.japan-guide.com/e/e2358_012.htmlhttp://www.japan-guide.com/e/e2358_002.htmlhttp://www.japan-guide.com/e/e2358_008.htmlAs for your plan, I would move things around a bit. I would go to Osaka first, and then visit Mount Koya. I would recommend staying in Namba in Osaka to begin with. Jet lag does different things to people, and the Minami may work on the first day.
http://www.japan-guide.com/e/e4001.htmlYou can get from KIX to Namba using the Nankai direct train or by bus.
Getting around a city, be it Osaka, Kyoto, or Tokyo can be done with an IC card for just a few trips, or a day pass. Tokyo might be the exception because of the JR Yamanote Loop, and in some places Osaka, but in general, if you plan to go to several locations, a day pass will run you less than 1000 yen. Here is a link to IC cards:
http://www.japan-guide.com/e/e2359_003.htmlI depends on what you have planned to do that day. If, for instance, you are at Namba in Osaka, and your plan is to go to Universal Studios, and spend the entire day there, Use the IC card to pay for JR-Namba Station to Universalcity Station, which will cost you 180 yen each way. You transportation cost is 360 yen. If, on the other hand, you plan to go to Osaka Aquarium, Osaka Castle, Umeda Sky Observatory, and Shitennogi Temple, then get a pass.
Some of this could be covered by the JR Pass, for instance, the round trip from JR-Namba Station to UniversalCity Station, so your 4000/day pass can cover a day which costs 360 yen, but other places, such as getting to Osaka Aquarium, will not be covered. You have to use the Chou subway line.
After Namba and Mount Moya, I think that a Jr Kansai Hiroshima Pass will get you to Hiroshima and back, to Kyoto, nara, and a side trip to Himeji or Kobe. There is also a possibility of going to a hot springs resort or onsen, such as Kinosaki Onsen, Kii-Katsuura, or Shirahara, before the pass expires. This pass is 13,000 yen for five days. Here is a link:
http://www.japan-guide.com/e/e2361_kansai_hiroshima.htmlThis still leaves getting to Kanazawa, Takayama, and Tokyo. I would work for Kanazawa and Takayama. There is the JR Takayama Hokuriku Area tourist pass, which will allow you to take the bus from Kanazawa to Takayama through Shirakawa-go (spend some time here). From here, you could take the Nohi bus from Takayama to Tokyo, or use the Pass to get to Nagoya, and get a discount Shinkansen ticket to Tokyo. The Nohi highway bus take about the same amount of time to get to Shinjuku in Tokyo, and you would not have to switch trains with your luggage twice. The bus is 6,690 yen:
https://www.nouhibus.co.jp/english/ This may not be the best option, if you have to get back to KIX for your flight. If this is the case, switch the Kansai Hokuriku and the Nohi bus for a Hokuriku Arch Pass, and use the Nohi bus system for a day trip from Kanazawa to Takayama. A final Link:
http://www.japan-guide.com/e/e2361_arch.htmlThis could take some time for you to digest. If you could tell me which airport that you plan to exit Japan from, and when you are going, I could be more specific.
