Take a JR train from Nagasaki to Isahaya Station (8 minutes, 1350 yen for an unreserved seat on the Nishikyushu Shinkansen or 30 minutes, 480 yen by local train). At Isahaya Station change to the Shimabara Railway (Shimatetsu) to Shimabara Station (70 minutes, 1460 yen, 1-2 trains/hour) or a Shimatetsu bus to Unzen Onsen (90 minutes, 1400 yen, hourly). The trains from Nagasaki to Isahaya are fully covered by the Japan Rail Pass, but the Shimatetsu trains and buses are not.
Alternatively, Nagasaki Kenei Bus operates infrequent direct buses between the Nagasaki Bus Terminal (across the street from Nagasaki Station, bus stop #3) and the Unzen Onsen Visitor Center. The one way trip takes about 100 minutes and costs 1850 yen.