Yes, about a week is enough to visit the places roughtly along the Transcanada Highway or an hour away from the main road. I was there about 8-9 days but I did drove from Port-aux-Basques to St. John's and back again, because the boat back from Argentia did not run anymore (late september), or the last boat did not fit in my schedule.
I visited Gros Morne (2 nights), Trinity (2 night), St. John's (2 nights), Twilingate (2 nights) and one night in Bangkok...uhm, sorry Port-aux-Basques on the way back.
Unfortunately I did not have time to go to Burgeo on the southcoast or to the northern Peninsula.
O, by the way, small car is enough, no 4wd needed on those routes I mentioned.