Hi,
Rail is by far the best. The airport has a subway station and it's only 5 minutes to reach Hakata Station, where you can get a bullet train to Hiroshima. If you are only going that distance, a 7 day Pass may not pay off for you. But you can still take advantage of a JR Hiroshima Yamaguchi Area Pass and save a lot of money. Plus, you can also ride the fastest bullet train, the Nozomi with it.
For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
Don't miss the Nanzoin Temple which is one of Japan's finest yet almost totally unknown to the outside world and completely FREE!
Look into a Fukuoka Tourist City Pass also.
You can also see a lot of previews on the best there is to see here.
For Hiroshima, you can see the city itself, but don't miss Miyajima, which is a real gem. Iwakuni is also nice, with its iconic historical bridge and mountaintop castle. Onomichi is famous for its temple tour and wildly ornate Kosanji Temple.
You'd need to pay a bit more for that since it is out of the pass's service area though.
Best of Luck!