Dazaifu is accessible via Futsukaichi (二日市) station on the Nishitetsu Tenjin-Omuta Line.
From Tenjin station in Fukuoka there are at least a dozen trains per hour operating to Futsukaichi. The journey is 12 minutes by Limited Express or 25 minutes by Local train, and currently costs ¥390.
If you want to head from JR Hakata Station to Dazaifu, the best option is to take a local bus to Nishitetsu Yakuin Station (¥100) and there take Nishitetsu train to Dazaifu (transfer at Nishitetsu Futsukaichi). Some local buses are available from JR Futsukaichi station to Dazaifu too, but they are rather expensive and infrequent.
Take the main road number 3 south out of Fukuoka, which passes through Dazaifu. This can be crushingly slow for such a short distance. A better option is to take the toll road south out of Fukuoka (you can access from various points around the city) and get off at the Dazaifu interchange, which will cost ¥600 as of May 2007.
Dazaifu has little of interest to the average tourist beyond Dazaifu Tenmangu Shrine and the adjacent Kyushu National Museum, however, if you should want to explore the remaining ruins, you are advised to walk or bicycle.
There are a variety of festivals held at various shrines across the town throughout the year. If it is possible, try to attend the monthly flea market in the grounds of Tenmangu Shrine, where kimonos can be had for ¥1000 or less.
Simply put, travel to Fukuoka city for shopping. Dazaifu houses merely the usual Japanese stores: Nitori, Uniqlo, GooDay, and so on.
There are a huge number of restaurants serving typical Japanese fare in the immediate proximity of Tenmangu Shrine.
Again, travel to Fukuoka city for evening entertainment.
Travel to Hakata via Futsukaichi, and from there to anywhere.