Hi there,
You'll be traveling quite a bit and I recommend using a 14 day RJ pass here.
Here's a quick breakdown of ticket prices without a JR Pass.
Tokyo - Nagano ¥ 8,000
Nagano - Shirakawago (assuming you go via Kanazawa and take the bus there) ¥ 8,760
Kanazawa - Kyoto (assuming you 1st take the bus back from Shirakawago to Kanazawa). ¥ 6,700
Kyoto - Osaka ¥ 560
Osaka - Hiroshima ¥ 10,030
Hiroshima - Osaka ¥ 10,030
Osaka - Nara ¥ 800
--Before I go on here, can you let me know for sure that you plan to do Tokyo - Nara - Tokyo at this point? It generally would not make sense to do this, unless you have a specific plan?
Though a quick glance at ticket prices so far certainly allows you to make good savings with a 14 day JR Pass.
Hope this helps,
Daniel