If you can cover all the spots you have mentioned on a twenty-one-day rail pass by JR Line trains, it will pay. Show you a possible breakdown, assuming you will be based in Shinjuku in Tokyo, Umeda in Osaka during your stay in Japan.
Currency/Yen
Narita Express, round trip between Narita Airport and Shinjuku: 6,220
Tokyo 23 Ward one day pass, 3 days' worth: 2,190
Shinjuku to Yokohama, return: 1,080
Shinjuku to Nikko on the bullet train inbetween, return: 9,640
Shinjuku to Lake Yamanaka, bus, one way: 2,000 (this isn't covered by the pass, so should be excluded)
Lake Yamanaka to Mishima, bus, one way: 2,300 (this isn't covered by the pass, so should be excluded)
Mishima to Nagoya, one way: 8,270
Nagoya to Takayama, one way: 6,070
Takayama to Osaka via Nagoya, one way: 10,610
Osaka to Kyoto, return: 1,080
Osaka to Himeji, return: 2,900
Osaka to Hiroshima, one way: 10,150
Hiroshima to Miyajimaguchi, return: 800
Hiroshima to Shinjuku, one way: 18,250
The amount is 77,260 yen, as opposed to 57,700 yen of 21-day JR pass. What is more, you'll be liberated from purchases of individual tickets on each trip, by which you can save on a lot of time and hassles. You could optimise the value much more if you make side trips on a whim, e.g. Hiroshima to Fukuoka, or Nagasaki.
For train connection and price search, use these sites.
http://www.jorudan.co.jp/english/ http://www.hyperdia.com/cgi-english/hyperWeb.cgiThe bottom line is that go for the pass.
Happy New Year!