Unless it's one of the largest cities in Canada, your airline choice will be limited to Air Canada (+United for connections) or Westjet. Good advice so far about researching ahead of time.
A great source for last minute flights is http://us.lastminute.com which is often cheaper than airlines because they bundle the flight together with a cheap car or hotel rental to qualify for a fare class that doesn't require advance purchase. http://www.smartertravel.com/last-minute/ is also very good, this one tracking the weekend and last-minute airfares that airlines announce during the off-season (ie, not much of these in summer or around holidays, but lots in winter). These 2 would be first place I checked, as well as http://www.bookingbuddy.com
Your task might seem daunting at first glance; but relax -- you can find the answer you seek in a few simple steps.
It's not so much a matter of which airline will give you the best price, it's a question of WHERE you find that price. The most efficient way to find that answer is through a "roll-up" site that searches multiple other sites for you simultaneously.
My personal favorite is Kayak.com. They search all the usual suspects like Expedia, Orbitz, CheapoAir and BookingBuddy; but also the airline's own sites, such as Continental.com. They will list ALL flights for you in order of price, beginning with the cheapest. In this way, they do all the work for you: It's like doing 100+ searches with one click of your mouse!
And no, I don't work for Kayak; I just love their site. I used to travel frequently with a large group, and we'd play a "game" in which the person who found the best fare would win a dinner compliments of the other players. I've had many, many free meals because of Kayak.
Congratulations on your soon-to-be new arrival, and happy travels!
With the baby not due till April, it may be cheaper to purchase a fully refundable or an open-return ticket now than to wait to buy a last minute traditional ticket later. Or check farecompare.com which can give you side-by-side comparisons of all airlines that fly between certain cities; keep in mind though that some of the flights you will see on farecompare.com will have connections but you can use their advanced search tools to limit the number of connections you want to take, and to see how you can save some money by flying on different days (for example, flying on Tuesdays is often cheaper than flying on Mondays because Monday is a peak travel time for business travellers).
Since you don't mention cities, it's hard to give you an answer. Try Westjet if they fly between the cities that you need. They are often less expensive.
In general, you will almost always pay the highest possible price if you wait until the last minute. Discounts normally have advance purchase requirements, minimum stay requirements, lots of restrictions on date changes, and no refunds. Flexibility costs money. Be prepared to get seriously gouged if you buy just before departure and there are no discount airlines in the market.
There is a list of great cheap travel sites at http://www.travelbargainsites.com One site compares all the airlines for the best fare. Once you find out the lowest going rate, then haggle with Priceline or Hotwire.