JRPass

Hi there!

A 14 day pass comes in at ¥45,100, while a 21 day pass is actually cheaper per day, and comes in at ¥57,700. To understand which option makes most sense for you lets take a look at your basic ticket costs (assuming Odawara for Hakone/Mt Fuji):

First 2 weeks travel

1. Narita airport terminal 1 > Kyoto: ¥15930, (238mins, 592km)
2. Kyoto > Nara: ¥690, (45mins, 41km)
3. Nara > Hiroshima: ¥10790, (167mins, 393km)
4. Hiroshima > Osaka: ¥10520, (109mins, 345km)
5. Osaka > Odawara: ¥12270, (159mins, 472km)
6. Odawara > Tokyo: ¥1450, (36mins, 83km)

TOTAL: ¥51650 for 6 legs

Full 3 weeks travel

1. Narita airport terminal 1 > Kyoto: ¥15930, (238mins, 592km)
2. Kyoto > Nara: ¥690, (45mins, 41km)
3. Nara > Hiroshima: ¥10790, (167mins, 393km)
4. Hiroshima > Osaka: ¥10520, (109mins, 345km)
5. Osaka > Odawara: ¥12270, (159mins, 472km)
6. Odawara > Tokyo: ¥1450, (36mins, 83km)
7. Tokyo > Nikko: ¥5430, (121mins, 150km)
8. Nikko > Tokyo: ¥5430, (117mins, 150km)
9. Tokyo > Kamakura: ¥1900, (48mins, 51km)
10. Kamakura > Tokyo: ¥890, (58mins, 53km)
11. Tokyo > Narita airport terminal 1: ¥2940, (58mins, 79km)

TOTAL: ¥68240 for 11 legs

As you can see, you will save a minimum of ¥6550 on the first 2 weeks with a 14 day Rail Pass, but if you opt for the 21 day pass your savings will be even greater at ¥10540. Because of this I recommend you go for the 21 day pass for your trip, and augment your Tokyo travel with a metro card as needed. While you are in Tokyo you can also get very good use out of the JR Pass as there is an extensive JR rail network across the city. You can refer to this map of JR lines in Tokyo as part of your planning to make even more savings.

Hope this helps!