Do yourself a favour and get out of the downtown core. It's nice and deserves to be seen for first-timers, but there's so much more to see and fortunately, the metro makes it all easily accessible (most of the time).
Get out at Jean-Talon and check out the Jean-Talon Market. It's better during the summer but it still has some great food year-round.
Get out at Place St-Henri and walk on Notre-Dame going East until you get to Atwater Market, another great market. The Notre-Dame strip is also home to many great food options. You can also reach the Lachine Canal there which is super nice for walks.
Get out at Laurier and go East to explore the Plateau and Mile End, probably the most trendy neighbourhoods in the city right now and home to many other great food spots, especially on Fairmount with Fairmount Bagels and Wilensky's if you're craving a bologna sandwich. St-Viateur has St-Viateur Bagels (both bagel spots deserve to be tried) and some nice italian coffee spots.
Get out at Sherbrooke and walk East to reach Parc Lafontaine, a nice place to take a walk, feels a bit like a small Central Park. You have La Banquise for poutine (although you will wait and I personally don't think it's that great) and Ma Poule Mouillée (portuguese chicken) right next to the park.
Get out at Pie-IX and check out the Olympic Park (Biodome, Insectarium, Planetarium, Olympic Stadium, Saputo Stadium, Botanical Gardens, etc.).
Get out at Mont-Royal and take the 11 bus up the mountain to the top of Mont-Royal for some of the best sights this city has to offer.
Of course you can also drive/take cabs to all these places but the metro is pretty simple to use and gets you to all those places fairly easily. Montreal is also a great walking city and there's no better way to get a feel for all the different neighbourhoods. Have fun!