I've been to Fukuoka a number of times, though not through Tokyo. All the same, I'd fly, unless you want the shinkansen experience. Since it's from Tokyo, it should be relatively cheap. There are discount airlines, like Skymark, that fly basically from Tokyo to everywhere in Japan for cheap. It's usually cheaper to buy well in advance, like with most flights.
In Fukuoka, I don't find the city all that exciting. I was always there for business, so I didn't have time to explore outside of Tenjin though. Tenjin is the downtown area. Fukuoka is famous for ramen and beautiful women, which is nice. There are a lot of nice onsen areas all around, so day trips would be a good idea.
Hopefully others here have more to tell you!