You can use JR West Japan's website.
Take note that the main station in Fukuoka (where I live) is not called "Fukuoka Station" but Hakata Station. So in the above search I entered "Hakata" in case that is where you intend to go to. If you really wish to go to Fukuoka Station (a non-JR line), just change the search or you can also take the 100-yen loop bus right in front of Hakata Station.
So in the above example, if you wish to arrive at Hakata by 12:00 tomorrow (Feb 6), then you would need to leave Hiroshima at 6:24 am, make 3 transfers, and pay a total of 6,380 yen if you are an adult.
I noticed that in this search result it shows that you have to take a "Reserved seating" for 1,050 yen. However, I checked another site (Navitime, in Japanese language only) and it shows an option for an unreserved seat for 520 yen only, lowering the total fare to 5,850 yen.
There is discount fare called "Seishun Juuhachi Kippu" (青春18きっぷ) available from JR each spring, summer, and winter. Last spring it was sold from Feb 20 to Apr 10 and was available to use from Mar 1 to Apr 10. There is no information yet for the coming spring, but I think that it will be around the same time. It's only about 12,000 yen but you can use it for 5 full days on local trains and even some ferries. Check this link for more info.