From Tokyo to Hiroshima to Fukuoka airport

Good day. I will be travelling with my family to Tokyo and Hiroshima in July/2014. The idea is to get a train from Tokyo to Hiroshima and then, at the end of our trip, we need to get another train from Hiroshima to Fukuoka Airport. Which one is the best choice we can have? Thank you!

Use Hyperdia for timetables and fares.

Will you fly into HND or NRT?

If you leave Tokyo / Shinagawa Station by Shinkansen, visit Hiroshima City and reach Hakata Station by Shinkansen all within 7 calendar days,
you can get issued a JR basic-fare ticket for the whole route to Hakata, which can be valid for 7 calendar days (and does not allow you to turn back or to skip any section on the route),
and the JR cost will be about 25000 yen per adult;
so, a 7-day Japan Rail Pass (29110 yen) is not likely to pay off unless you make other trips by JR off the route between Tokyo / Shinagawa and Hakata.
I do not think of any other rail road pass which can be more cost-saving.

Your transportation options between Tokyo and Hiroshima are explained in the following article.
- japan-guide.com: How to travel between Hiroshima and Tokyo
http://www.japan-guide.com/e/e3404.html

You don't write anything about what you want to go see, nor the length of your trip - so it is hard to give something definite.
But since you can see Hiroshima and Miyajima in about 2 days, there is a lot of potential for you to travel around for the rest of your time, and a 7 day JR Pass would serve you nicely. There are many nearby sites you can see, including Onomichi, Kurashiki, Iwakuni, and the Yamaguchi Akiyoshido limestone caves, which are some of the world's finest.
Save a bit of time to see some of Fukuoka as well - it is a nice laid-back city with lots to see and do. You can get some idea of good places on:
https://www.youtube.com/playlist?list=PL7LcPcmoSg-j4RLa6iC7ayvBAbOn8-0...