Either WestJet or Air Canada are good for a cross Canada flight, they are the major players in Canada. There is no OW domestic airline, and I really wouldn't recommend going back and forth to the USA simply to use AA. It adds a lot of time and expense, plus hassle at the border each way. You also will not be able to book a Canada-USA-Canada ticket as one ticket without a stopover. Even with a stopover I have heard of people being denied boarding on those itineraries due to cabatoge rules. No 261 compensation either, you're just plain SOL.
Depending upon your budget and what you want to see, I would recommend taking Via Rail's The Canadian. It really is an excellent way to see the country, plus gain an appreciation for how vast Canada really is. Be aware that the rail network in Canada is extraordinarily busy now, so don't plan a same day connection at your destination (Vancouver). The time keeping has improved with a new schedule, but previously 12+ hours delays were the norm.