In Montreal for breweries go to Dieu du Ciel and Benelux. For food try Shwartzes on St Laurent for smoke meet (you could also go to The Main across the street just as good and less buisy). Au Pied du Cuchon and Joe Beef are amazing but can be pricy but not as pricey as you would expect for world class chefs.
The fine arts musuem is great, but my favorit is the Redpath musuem inside McGill it is free and has dinosaurs.
For Poutine, there is no way around it: La Banquise. Located east of downtown near Parc Lafontaine at 994 Rue Rachel E, Montréal, QC H2J 2J3 , it is a legendary 24 hours poutine place. They have 30 kinds of poutine and all the beers they serve are from local microbreweries. For poutine, I have a legendary appetite and I can barely finish the "large" size. So stick to the normal size and enjoy!
http://labanquise.com/en/poutine-menu.php . My favorite is the "Fred Caillou".
In addition to Poutine, the signature Montreal steak cut is the "Rib Steak" topped with roasted garlic and Smoked Meat spices. In the US, it would be a bone-in Ribeye and for a fine steak in Montreal, I strongly suggest Moishes. Its near Schwarz on St-Laurent street and they have an amazing after 9pm menu.
http://moishes.ca/x72P578w/wp-content/uploads/2018/01/Moishes-menu-after9-minuit-2017-20-dec.pdf
Are you renting a car? Most Costco around Montreal will sell you two 50$ certificates to Moishes for 80$. That would represent a nice discount as I figure two people on the After 9 menu willl come close to 100$ by the time you count the tip, drinks and taxes. Moishes also has their own parking, a rarity in that part of town.
If you have a car around Montreal, right near the airport is my favorite Smoke Meat Place. It is called Delibee and is a true hole in the wall restaurant, closed on sundays and attached to an old school tavern. The address is 24 Av de la Baie de Valois, Pointe-Claire, QC H9R 4B3 . For me, the Smoked Meat is much better than Schwarz and their fries are delicious. The cole slaw is served with a spicy vinegar and I took a die-hard Schwarz fan here once. His verdict? As good as Schwarz, which even he did not think possible. Give them a call before at (514) 693-1163 to make sure they're open when you want to go. Tip: the rib steak is AMAZING and they even serve it with a little starter of smoked meat and roasted beef sausage.
For a day trip, I have a suggestion if you like ciders or wine in addition to beer: head to the eastern townships. You can visit a lot of wineries and cideries that offer tasting and you can find some amazing things there. Ice Cider is incredible.
Might I suggest a way out of the way restaurant if you can drive? It is not touristy at all and they did not have an english menu last time I was there with a friend, but the food was out of this world. The restaurant is inside an ancestral home and the French cuisine is as good as what you would find in a good restaurant in France. Prices are incredible too: 49$ for a five course meal featuring high french gastronomy. The website is
http://www.lefolichon.com/topic/index.html and send me a PM if you want to try it. It is located in my city.