Travel between Tokyo, Kyoto, Osaka and Hiroshima

We will be arriving and staying in Tokyo for 3 days perhaps doing at least one day trip to Mt Fuji whilst there. We are then planning to travel from Tokyo to Magome, then to Kyoto where we will stay 3 or 4 nights before travelling to Kii Tanabe. This will take approx. one week.

From Kii Tanabe we will be hiking the Kumano Kodo walk for another week.

In the 3rd week we plan to travel from Kii Katsura to Osaka staying in Osaka for a few nights and visiting Himeji Castle, Hiroshima and Miyajima.

I would like to purchase a JR ticket for the first and third week - NOT the second week.
Is this possible and is it a cost effective option.
Dilhara