I am going to second seeing the Oratory of St. Joseph. It was begun, as mentioned, by St. Andre, and has a good view of the city, since it is built on Mont Royal. The basillica itself is enormous, but if you do go, make sure you see St. Andre's original little chapel, and his bedroom up above it. Some people miss it, because it is up the side of the mountain to the right of the basillica. The basillica itself is rather modern in decor on the inside, but the size of it is rather breath-taking. The original chapel has the more traditional charm. St. Andre's relics are kept in the basillica, so you can stop and say a prayer. There is also a huge room full of votive candles where many, many canes were left behind, attesting to cures atributed to the intercession of St. Andre. The heat from the candles is intense, but it is beautiful, as the hundreds of candles go up and up stairs. You can pick up a brochure there with directions on how to lead a self-guided tour, as there is a decent amount to see. They do have a museum with Nativity sets/chreches from around the world, and if you can make it in time to hear the carillion, it is lovely.
I have to admit, I find it somewhat stressful driving in Montreal itself. It is hectic, and very confusing for those who are not familiar with it. There is lots of merging, and often lots of construction detours, and you have to think very quickly. As, I suppose, in any large city there are also drivers who can be rather aggressive. It helps if you are fluent in French; if not, a gps or someone familiar with the area is invaluable
If you can handle it, though, it's worth it, of course
Quebec itself is filled with many, many churches. It is said if you throw a pebble in any direction, you will hit a church
However, the other two main national shrines are Notre-Dame-Du-Cap (Our Lady of the Cape) which is in Trois Rivieres and St. Anne-de-Beaupre in the city of the same name. Our Lady of the Cape is my personal favorite, perhaps because it has the most peaceful and reflective atmosphere
The original chapel with the original miraculous statue is beautiful, and there are often candle-lit processions on the beautiful grounds around the pond after dark. St. Anne's is pretty neat, there is a LOT of history, and a good amount to see in the basillica. It actually has a LOT of relics, scattered throughout the Church, and the "chapels", and the altars under the church. There is also the "scala santa" up on the building on the hill, which is kind of a nice pilgrimage-y thing to do.
If you do have a car, and don't mind driving a few hours, then I would DEFINITELY suggest Old Quebec. It WILL be crazy-insane there July 5th to 15th because of Festival d'été de Québec (Bon Jovi, Aerosmith, Sarah McLachlan and many more famous artists will be performing). But, if you can handle it, or if you can make it outside those days, it is AMAZING. The United Nations actually declared Old Quebec City a World Heritage Centre. See here: http://whc.unesco.org/en/list/300
. On the outskirts are the Plains of Abraham, where a famous battle was fought between the French and English. The city itself still has its original ramparts and bastions, cannons, and cobblestone streets. There is nothing else in North America that has been preserved this way. The historic flavour of the city is tremendous. There are gorgeous gold-leafed historic churches, and many awesome original buildings, turned tourist. The Chateau Frontenac is an extremely famous historic hotel (but be forewarned, it is also very expensive--several hundreds for their cheapest room. Royalty often stayed there in the past. It has all the modern conveniences, now, of course!). Also, there are horse drawn carriage rides for hire, and nightly lineups of street performances in the square. Buskers are usually scattered throughout the city as well. There is even an old tree that grew around an original cannon ball! There is SO MUCH to see. There are lots of shops for souvenirs, and lots of places to eat. Some places to eat can be a little pricey, depending where you go. They usually post their menues, outside, however. If you want to do fast food, there is a McDonald's, too. Old Quebec WILL be busy, but it is totally worth it. If you want to see it, keep in mind that you can't take a car through it on the original streets, so wear comfortable shoes. Some of the city is very steep. Keep in mind that it was first discovered in the early 17th century!