Tosu is the intersecting station between the JR Kagoshima Line and JR Nagasaki Line.
A plethora of limited express trains operate between Hakata station in Fukuoka and Tosu. If you're desperate to get to Tosu quickly, a limited express train will get you there in about 20 minutes at a cost of ¥1440. There are about 5 or 6 limited express departures each hour. A Japan Rail Pass allows you to take a nonreserved seat on these services at no charge... just hop on the train. Or, you can reserve a seat, also at no charge, if you feel inclined to do so.
Otherwise, you're better off taking a regular service from Hakata which costs ¥540. Rapid trains reach Tosu in 40 minutes; local trains take one hour. In either case, your train will probably pull over to let a limited express pass.
From Nagasaki, Tosu can be reached by limited express in 1 3/4 hours (¥4090), or by regular train in 2 3/4 hours (¥2420).
From Kagoshima, Tosu can be reached by the Tsubame service (Tsubame Shinkansen + Relay Tsubame Limited Express) in about 2 hours (¥9420).
All of these rides are at no charge with the Japan Rail Pass.