Hi Kosmo,
You can you visit Kyoto - Osaka - Hiroshima in any order you wish. Hiroshima is the most south so you may want to plan this between your visit to Kyoto and Osaka to spread your travel around.
I would recommend visiting the following places in each places:
Kyoto:
Consider a visit to Kyomizu Dera, Nishiki Market and a hike around Kurama if you have the time.
Osaka:
For Osaka, take a look on this website which lists (almost) all the interesting places in Osaka, which are sorted by interest. My personal recommendation would be a visit to Shinsai Bashi..
In Hiroshima, make a day trip to Miyajima for sure. This is one of the most symbolic places of Japan and not to be missed on any trip. I would also recommend a visit to the Peace Memorial Museum which is an import part of history.
As for a visit to Fukuoka or an other city in the south, once you are in Hiroshima it is only a little more down to Fukuoka or Kyushu. So you could certainly visit it but do remember that you have to be back in Tokyo by day 7. As for going up north beyond Tokyo, Hokkaido is pretty far away. A trip Osaka - Sapporo takes ~14 hours of straight train travel so that may be a tickle to far for just a week of travel. You could go up to Sendai but much more further may just be too far away.
I hope this helps!
Daniel