Hi there!
Lets take a look at the ticket prices of your itinerary to get an idea for the Rail Pass options that will come in cheaper:
- Tokyo > Himeji: ¥15210, (220mins, 644km)
- Himeji > Hiroshima: ¥8070, (60mins, 249km)
- Hiroshima > Miyajimaguchi: ¥400, (26mins, 21km)
- Miyajimaguchi > Iwakuni: ¥320, (23mins, 19km)
- Iwakuni > Kagoshima: ¥17830, (272mins, 614km)
- Kagoshima > Aso: ¥17830, (272mins, 614km)
- Aso > Tokyo: ¥17830, (272mins, 614km)
TOTAL: ¥77490 for 7 legs
As the prices of the 7, 14 & 21 day Japan Rail Passes are ¥28,300, ¥45,100 and ¥57,700 respectively, you will save money on your trip using any of the Japan Rail Pass types.
Often local trains will not have First Class seating, however you will be able to use First Class/Green for many of the longer legs you list. Some JR-West Hikari Bullet services do not have Green Car seating, so make sure you use the Sakura Bullet service if you decide to go this way (and reserve in advance).
Hope this helps!