Assuming you'll have a car for the two weeks, make your way over to the other side of the island to see Gros Morne National Park. It's a seven hour drive and you can pit stop in Gander for the the aviation museum. Two weeks will let you see plenty. If you haven't booked your flights yet, I would suggest going around the time of the Regatta, North America's oldest sporting event. It is scheduled on the first Wednesday of August, but plan for a few days in St John's around then since if it rains they postpone to the next day. It is not the fanciest of events, but gives you a look into the heart and soul of that wonderful province. However, late spring and early summer is best for viewing icebergs.
My favourite activity in St John's was walk up to Signal Hill from downtown, and on the return took the trail down along the coast and through the mouth of the harbour. You exit through the Battery district.