JR worth it for us?

Hello,

We hope someone is kind enough to answer our question:
We already did some research but are still unsure if we should get a rail pass for our trip in Japan. We would like advice on what would be the cheapest option to travel.

25-4-2014 Osaka
26-4-2014 Osaka>Koyasan
27-4-2014 Koyasan>Nara
28-4-2014 Nara>Kyoto
29-4-2014 Kyoto
30-4-2014 Kyoto
1-5-2014 Kyoto
2-5-2014 Kyoto>Tokyo
3-5-2014 Tokyo
4-5-2014 Tokyo
5-5-2014 Tokyo
6-5-2014 Tokyo
7-5-2014 Tokyo>Osaka
8-5-2014 Osaka>Arima>Osaka
9-5-2014 Osaka

What would you advise us?

Thanks very much in advance,

Johan and Asha