I'm going to give you some low cost passes to consider for your trip. You could get the Seishun 18 Kippo pass for a supplement.
Here we go:
For Nikko, look into one of the Tobu Nikko Passes. There are several flavors of these passes, and some have park entry fees bundled in with the pass:
http://www.japan-guide.com/e/e2358_002.htmlYou could take local trains on the Seishun 19 Kippo pass to get you to Matsumoto, and then take the Nohi bus to Takayama. I have gone to the Hyperdia site, and plugged in Shinjuku to Matsumoto, and I disabled Airplanes, Shinkansen, Private trains, and Limited Express trains, to give you an idea of what using a Seishun 19 Kippo pass is like. It takes 4 1/2 to 5 hours.
http://www.hyperdia.com/en/cgi/en/search.html?dep_node=SHINJUKU(JR)&ar...I wanted you to know the tradeoff of time -vs- money.
Instead of taking a train to Mastumoto, there is a direct bus to Takayama from Shinjuku. The bus is a hour longer than the Shinkansen/Limited Express train route, and will be a lot faster than riding many local trains. The bus is run by the Nohi bus line, which also has busses from Takayama to Matsumoto, and Kanazawa through Shirakawa-go.
https://www.nouhibus.co.jp/english/For getting to Kanazawa, and Kyoto, there is a very good bus/train pass called the Takayama-Hokuriku Area Pass:
http://www.japan-guide.com/e/e2361_takayama_hokuriku.htmlYou may want to use the last day of this pass, if you have one, to go to Osaka, and then pay the rest of the way to Himeji. This could save some money.
Since you will be in Kyoto on the 7th, you might want to go to the Tanabata Festival. Here is a link:
http://www.deepkyoto.com/kyoto-tanabata-festival/At the end of your time in Kyoto, you might get the Wide version of the Kintetsu rail pass. Use it to do a day trip to Nara. On the following day, you will use it to go to Namba in Osaka, and use one of the airport ticket from the Kansai Pass to get to KIX.
http://www.japan-guide.com/e/e2357_006.htmlFrom there, take a train to Wakayama to catch the ferry to get to Takamatsu:
http://www.japan-guide.com/e/e2355_01.htmlYour return trip is Ferry to Wakayama, train or bus to the airport,
a another airport ticket to get to Namba, and then a Kansai limited express train to Nagoya. If you would rather do hiking around the Kumano Region, you could also look into the Ise-Kumano-Wakayama Area Tourist Pass:
http://www.japan-guide.com/e/e2361_ise_kumano.htmlA link to the Kumano Hiking trails:
http://www.japan-guide.com/e/e4950.htmlThis pass will also get you to Nagoya. So I would use it instead of the Kintetsu pass.
I really suggest that you use Hyperdia to figure out the cost of different train legs using local trains, and not the Shinkansen and Limited express. This is a serious amount of time that your will be using.
I hope this helps.