In Quebec City is the old Frontenac walled city, an absolute pleasure to wander and beside they have the oldest military base in Canada. I recommend spending a full day down in that part if you can. If you're brave take the bazillion stairs down from the plains de Abraham and see how jellied your legs get along the bottom. Edit: they also have an amazing music festival in the summer, check if it's on when you're there!!!
Ammolite jewelry, native American crafts (dream catchers and soapstone carvings (the inukshuk are a safe bet) would probably be your go to- unless the RCMP moose/beaver stuffed animals would be appreciated. East coast seafood is pretty famous to us on the Western end of the country, PEI is famous for their potatoes and red soil, alberta for beef, and British Columbia is called bring cash for a few different reasons.
Respect the wildlife. Here it really is wild, and that moose/elk/mountain goat will take you out if it wants to.
If you know someone who appreciates it icewine from Niagra falls area, or Newfoundland screech for the daring. Try Crown Royal if you drink whiskey or enjoy a good rye. Try second cup coffee at least once, it's the strongest coffee by caffeine content in Canada, and Tim Hortons can keep their swill. Just because it was started by a famous Canadian hockey player don't mean it stayed good when the franchise was sold to an American conglomerate.
Check out the small fusion restaurants, and especially at least one small town Chinese restaurant- they have an interesting history in Canada and the food's as genuinely Canadian as any. Ask the locals about the best place to eat. And although winter sports are usually a bigger deal try to take in a basketball game(was created by a Canadian!), Lacrosse game, or spend at least one day at a national heritage site.