Hi there,
Lets quickly get a rundown on the cost of travel to your locations:
-
Narita airport terminal 1 > Kumamoto: ¥27,350 (486mins, 1372km)
-
Kumamoto > Nagasaki(nagasaki): ¥7,370 (124mins, 212km)
-
Nagasaki(nagasaki) > Hiroshima: ¥12,250 (198mins, 434km)
-
Hiroshima > Osaka: ¥10,820 (109mins, 345km)
-
Osaka > Kyoto: ¥540 (29mins, 42km)
-
Kyoto > Nara: ¥690 (44mins, 41km)
-
Nara > Odawara: ¥12,070 (188mins, 471km)
-
Odawara > Tokyo: ¥1,450 (36mins, 83km)
JR ticket cost: ¥72,540 for 8 legs
Based on your basic travel above, you'll make a minimum saving of ¥14,840 with the 21 day JR Pass, and also have the flexibility to add further JR travel to your trip at no extra cost. Because of this I can certainly recommend the 21 day JR Pass.
Additional locations
Given the length of your trip I also recommend taking a look at the following extras:
- Ibusuki hot sand baths at the bottom of Kyushu (overnight)
- Sakurajima island (overnight)
- Mt Aso (day trip)
- Kurokawa Onsen (overnight)
- Miyajima (overnight)
- Naoshima (overnight)
- Koyasan (overnight)
- Kinosaki Onsen (overnight)
- Arashiyama (day trip)
- Shingu / Kumano (overnight/day trip)
- Takayama (overnight)
- Kanazawa (overnight)
- Hakone Onsen (overnight)
- Nikko (day trip)
Many of these are accessible by JR but Kurokawa, Hakone, Koyasan, Sakurajima and Naoshima all require small extra charges for non-JR travel components.
Timings
Regarding timings, I think I'd aim to spend 4-5 days in Tokyo, 4-5 days in Kyoto/Osaka/Nara, possibly 1-2 days in Hiroshima, and then split the remainder of your time depending on your itinerary. Time here very much depends on your pace and preferences, so will depend on how many of the above extras you think you will do.
Beaches
There are many beaches across Japan, expecially in Kyushu/south - we have a good blog post on this with more, but you may want to investigate the area around Ibusuki as it is the most southerly and tropical of the areas you'll go on your trip.
Acommodation
You'll find that accommodation for a single female is very safe and is similar (if not more safe) to other western countries so there is no specific advice needed here I think.
Finally, I strongly recommend you purchase the book Japan by Rail. This guide has lots of great sidetrips and excellent detail on places to stay and so on.
Hope this helps!