If you start off at Hakata train station, you can take the　Kamome limited express train to Nagasaki which will take you about 2 hours and cost 4,710 yen.
Another possible way would be starting at Hakata bus terminal and take the highway bus (Kyushu Go) going to Nagasaki Eki Mae. This will take you 2.5 hours and cost 2,500 yen.

You can use the following sites to check the train connections and costs
http://www.jorudan.co.jp/english/
http://www.hyperdia.com/