JR covers nearly all of the major and medium cities across Japan, so no problem.
Hiroshima has many good places - within the city of course is the Peace Park/Museum, Shukkeien Garden and Castle, and the main attraction nearby is Miyajima. If that is still not enough, there are many places between or close to Hiroshima. Just a bit past Miyajima is Iwakuni with its iconic historical bridge and mountaintop castle. Onomichi is famous for its temple tour and wildly ornate Kosanji Temple. Nearby Kurashiki is a famous own city for its canals, and Okayama also has its own castle and one of Japan's other Top 3 gardens, Korakuen. If your kids are itching to explore, then in Kagawa there is the Konpirasan Shrine in Kotohira, with a marvelous paved mountain hike.
Kitakyushu is mainly an industrial city though, with limited sightseeing options. Kokura Castle is very nice, and the Moji Retro area is like walking back in time. Every year from late may the wisteria bloom in the Kawachi Fujien which is spectacular.
The walk through the Kanmon Underwater Tunnel is also pretty cool. There is one secret gem in the city though - the day and especially night view from Sarakurayama is just mind blowing.
But overall you might enjoy Hakata or Fukuoka City more. For Fukuoka, there are many open parks, shopping, seaside historical areas, and urban neon. See this site as a good guide.
For Fukuoka, 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.