Public transportation in Montreal is clean, safe and efficient. It's also easier than trying to find a parking place (it's very difficult to find a cheap parking place in downtown and almost impossible in old Montreal); all the important sites to visit are near a metro station anyway (like the slogan says : "It's smarter to take the bus/metro"). The corporation that manage the public transit system on the island is called the STM , it stands for Société des transports de Montréal (or the Montreal Transit Corporation if you prefer). You can find all the information you need on their web site www.stm.info, there are bus and metro schedules and maps, fare information and trip calculator (which is also accessible on Google Maps ).
If you have to go off the island, both the south shore and Laval (a suburb north of Montreal) have their own public transit systems. The RTL for the south shore and the STL for Laval. The south shore bus terminal is at Bonaventure metro station and the Laval bus terminal is at Henri-Bourassa metro station. The only metro station on the south shore is the Longueuil metro station. There are 3 stations in Laval. All of these transit commissions are supervised by the AMT who also manage the commuter trains.
Here are all the details on public transit in Montreal: