One of your first stops when you arrive in Montreal should be at the Info-Tourism office where you can find free tourist guides, informational booklets and people to help you plan your visit. You can also find information about current events and activities. Its address is 1001 Square Dorchester, and the local phone number is (514)873-2015. It's in the Dominion Square Building (the building's interior is worth a look, it has hand painted wooden ceilings) near the corner of Metcalfe street. You can get there via the Peel metro station (exit on Peel and walk down to Ste-Catherine). It's in front of Dorchester Square (it used to be known as the Dominion square), but you can enter through the Ste-Catherine entrance and go to the back of the building where the office is located.
Here is the most important thing you should get to have some free information about things to do : the Mirror and Hour newspapers (in English) and the Voir newspaper (in French) are three free newspapers. You can find them at most metro stations, bars, restaurants and newspaper stands. They have listings of activities (movies, theater, music, special events) for the week. You should pick one up when you arrive in Montreal but they come out on Thursdays and by the following Mondays, they are usually all gone. If you are not sure where to look for them, there's a big pile of them at the McGill metro station (near the The Bay store) or at the Place-Des-Arts in front of the coffee shop in the underground plaza. The Mirror has the most complete listing of activities and bars, the Voir and the Hour both have the same information on activities.
Another thing you should watch for : Info-columns, these are big columns on the sidewalk with maps of the district and info on things to see all over Montreal, they are easy to spot (they are usually either round, triangular or flat columns).
A good thing to do is to ask for a transit map at any subway station. It is a complete map of Montreal (but without a street index), with all the bus lines and subway lines, it also shows the entrances to the subway stations and the underground passageways.
For more sources of free information, try the Where can I get more (free) information before coming to Montreal? section.