By train
Kurume is on the JR Kagoshima Main Line.
From Hakata station in Fukuoka there are generally one local and two rapid trains per hour operating to Kurume, with more trains during rush hours. Rapid trains take 40 minutes, and local trains take about an hour, at the same cost of ¥720.
Paying a ¥600 surcharge ¥900 for a reserved seat will allow you to travel on one of the many limited express trains that operate on the Kagoshima line, which will bring you from Fukuoka to Kurume in half an hour. These trains operate on the same or higher frequency than regular trains, and include the Relay Tsubame for Kagoshima connecting to the Kyushu Shinkansen, the Ariake for Kumamoto and Hikarinomori, and the Yufu trains for Yufuin, Oita and Beppu.
If you have a Japan Rail Pass you can take any of these limited express trains at no extra charge; for unreserved seating, simply show your pass to the conductor on the train.
Kurume is also on served by Nishitetsu Railway on the Tenjin Omuta Line, stopping at Nishitetsu Kurume station which is about 2.5 km east of the JR station. From Nishitetsu Fukuoka station in the Tenjin area of Fukuoka, take any Omuta-bound train. Train tickets can be purchased from the ticket machines at the station, and cost ¥600. Limited Express trains depart every 30 minutes on the hour and half-hour, and take approximately 30 minutes.
By bus
Highway buses bound for Nishitetesu kurume station can be boarded at Fukuoka airport and run two or three times an hour until 10 at night. One can also board Kurume-bound highway buses at Nishitetsu Tenjin Station or JR Hakata Station. Buses take about an hour to reach Kurume.
By car
If you are driving to Kurume, you will be able to park your car in one of roughly a dozen cheap parking lots, a happy side effect to the Japanese economic depression. Look for parking signs near either of the stations. Expect to pay around 100 yen per hour.