There are five IFR airports on the Island - Deer Lake, St. Anthony and Stephenville in western Newfoundland, Gander in the middle and St. John's on the east coast. Goose is on the mainland. There are a few VFR airports around, Clarenville, Harbour Grace, Springdale etc.
They have flight schools in both Gander and St. John's where you could rent an airplane. PM me, and I'll see if I can put you in touch with one of the guys who used to post here on proon under the name of code blue. He'll be able to steer you in the right direction.