I have the following itinerary for our trip. Is the JR pass worth the expense?
March 18th Arrive in Tokyo
19th Tokyo- tour tsukiji fish market, sensoji temple, meiji shrine, shibuya crossing, shinjukan gyoen national garden, akihabara (Maid Café)??
20st Tokyo-
21nd Tokyo- Kamakura- Day trip
22rd Tokyo- Nikko- Day trip
23th Tokyo- Mt Fuji and Lake- Spend night in Hakone
24th Osaka – Sumo Wrestling
25th Osaka – Hiroshima-
26th Osaka- Koya-san
27th Osaka- Himji Castle
28th Osaka – Nara-
29th Kyoto- Kanasawa-
30t-Kyoto- Nanzen-Ji, Kiyomizu Dera, Sanjusangendo, Gin-Kakuji and Kin-Kakuji
31 - Kyoto-
1nd – Kyoto Geisha Performance
2rd Kyoto - Fly from Osaka
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Comments
In your case, yes, a 14 day pass would pay off since you are taking some long distance day trips in addition to the one way trip from Tokyo to Kansai.
A JR Pass would not get you all the way to the Mt Fuji area or Koyasan however, so you should look at some Odakyu passes and the Koyasan World Heritage Ticket.
http://www.odakyu.jp/english/deels/
http://www.japan-guide.com/e/e2358_012.html
You can also use the JR Pass for the Loop Bus in Hiroshima City.
You can see both Hiroshima and Miyajima with an early morning start. You can cover the Peace Park/Museum, castle, Shukkeien Garden and Okonomiyaki-mura, and then go down to take the JR ferry to the island. Aside from the Itsukushima Shrine with the giant "floating" torii, the Daishoin Temple is also a must-see place. There are several other pagodas and temples, and climbing Mt. Misen offers a fantastic view over the island and whole area.
If you have time, just a bit past Miyajima is Iwakuni with its iconic historical bridge and mountaintop castle.
For Kyoto it looks like you are going to 2 of the big 3 sights there - you can also see the Fushimi Inari Shrine if you want - it is open 24/7 so it's easy to fit in the early morning or evening and also avoid the crowds.
If you get the 14 day pass, it still leaves 2 days of paying out of pocket.
If you land at Narita, there is the 1000 yen bus:
http://accessnarita.jp/en/home/
and the 2 or 3 day subway pass can save money too.
http://www.tokyometro.jp/en/ticket/value/travel/index.html#anc03
For Kamakura, don't miss Enoshima as well. An Odakyu pass can help there too.
http://www.odakyu.jp/english/deels/freepass/enoshima_kamakura/
For Himeji, don't miss the Kokoen Garden next to the castle. To Make it a full day, go up to Mt. Shosha to see the huge temple complex there.
You can see previews of some of the best places to see in Japan on:
https://www.youtube.com/user/thejapanfaq/videos
To get to Hakone from the other side of the mountain (the 5 lakes), this site has more info:
http://www.japan-guide.com/e/e5206.html?aFROM=5200_6900