NF is a very large province. You can drive it (coast-to-coast!) in a full day but you won't see much on the Trans Canada (except moose!). I suggest you focus on either the east (St. John's-YYT) or the west (Deer Lake-YDF) coasts. You will want to rent a car to tour:
-West - Gros Morne, hiking, fiords;
-East - Terra Nova, outports on the Burin or Avalon Peninsulas.
Each has its +/- so the choice depends on personal preferences.
Finally, an "out there" suggestion. Fly into YYT, drive to Fortune on the Burin, and take the ferry to St. Pierre & Miquelon, France! Experience of a lifetime!