I'm going to second (or third or forth), Canada is huge. Second largest country in the world (though most of our population is within a few hundred KM of the US border - as is most of our infrastructure and highways).
If you only have 18 or 19 days, there is no way to make it across the country and actually see the country (with time to stop to take photos). It's just too big. Driving through New Brunswick, Quebec and Northern Ontario is a very slow go because of the Appalachians and then the Canadian Shield Though beautiful, they are not made for nice straight highways. That changes between Manitoba to Calgary, Alberta where the roads straighten out(but again, going through the Rockies in Alberta / B.C. is beautiful, but slow).
I've done the drive in pieces, from Halifax to Toronto several times (Law School) and from Toronto to Vancouver (once through the US, once through Canada). It is a LONG drive. Halifax to Toronto as straight shot is about 20 hours. Toronto to Vancouver as a straight shot is 3 or 4 times longer (if you drive through Canada - 2 to 3 times longer if you go through the US). I've also done the trip with the intent of documenting my journey across the Trans-Canada (which at times, actually most of the way, is little more than a two lane highway). It took 4 days from Halifax to Toronto and 9 days from Toronto to Vancouver. This didn't include any side trips at all. None.
My best advice as someone who has done this, and who has lived in Ontario, Nova Scotia and Alberta, would be to pick a section of the country and do it right. Spend your time seeing it, not driving it.
Briefly, to help you make a decision, the Maritimes are fishing villages and rough ocean front. You could see Cape Breton which is more like the highlands in Scotland than anything else. Southern Nova Scotia where there are still fishing villages down the Digby neck, the Bay of Fundy, with the largest tide in the world. If you like to camp, Kegi is one of our nicest parks. There is a lot of tourist places that are worth a look, but not a stay, some have mentioned Peggy's and Lunnenberg. There are authentic places to see as well. If you're really adventurous, you could try to arrange a trip to Sable Island (one of Canada's best kept wildlife secrets) or to Newfoundland / Labrador. Newfoundland and it's landscape is closer to the Norwegian Fjords than most would expect. Labrador is almost empty wilderness, expensive to see properly, but an untouched gem with never ending forests (think Siberia).
I'll leave Quebec out as I've only really spent time in Montreal and Quebec City. Montreal and Quebec are talked about because of their European feel, if you're coming from Europe this won't be all that special to you. Otherwise, aside from great restaurants and cafes (Montreal does have incredible bagels and sandwiches - see Schwartz's).
Ontario is also really really big. Toronto is a big city. It has a lot of great neighbourhoods, but it is a big city. We have our share of museums and art galleries, but think about what you want to see, the country or a big concrete city. Ottawa is also very nice, lots of green space, gorgeous parliament buildings, supreme court, great art galleries, a top notch military museum. Make sure you go that the Rainbow Room if you like jazz. If you want nature, head to Algonquin. Massive national park with never ending back country.
The drive around the top of the great lakes is amazing. There are a lot of beautiful places to stop for hikes. It is interesting to leave Toronto and then be in the middle of almost nowhere (punctuated by a few urban centres) until you hit Winnipeg.
Manitoba has Churchill and the polar bears, but you need to fly there. No roads. Incredible, but your timing may not make it worthwhile for the expense.
Saskatchewan and Alberta (until Calgary) are almost perfectly flat (Sask does have a number of beautiful river valleys). Take time to look at all of the grain elevators. Gorgeous.
Calgary is like the other big cities. And forgive me, without the charm. Unless you're there in July for Stampede of course.
Alberta west and north of Calgary is more like what you think of when you think of the big US parks. Mountains, wildlife, lakes. Beautiful.
BC is much the same, especially outside Van (again, lacking in charm - like Toronto. I won't ask for forgiveness here as those from Van love to talk about it). Northern BC is for going to see the grizzlies as is Vancouver island (a distinct place from the city of Van). Van Island is amazing. Again, think almost untouched wilderness. Surfing, hiking, cottages on the Pacific and big bears.
Okay, I was having fun. Sorry for the run on. Kinda reliving past trips. Once you've decided what you'd like to see, feel free to drop me a line and I can provide more specific tips on each part. I'd personally recommend the Martimes. You''ll already be in QC and its a short and entertaining drive out. That'll give you a few days to drive around Cape Breton, NS and perhaps get to Labrador or Sable if you're looking for adventure.
The Hopewell rocks and Shediac Beach are worth a visit. Heading back, the Gaspe Bay is certainly worth the diversion and if so inclined, there is a very easy hike up to the top of Mount Jacques Cartier with nice views.
