The Montreal Metro system includes the subway trains and road buses.
These offer excellent ways to explore the city of Montreal.
STM home page has fares and routes and schedules information.
Exploring Montreal is quite easy using the metro system. The subway trains are good to go someplace fast, and then take a bus to enjoy the street views without exerting much energy. Then, come back to visit the places and restaurants that look interesting.
Astonishingly, the buses manage to keep close to the published schedule. Never seen this happen in any other major city! Some routes at peak hours do miss schedules, given the traffic congestion. Though most times you can depend on the schedule. And there are many smartphone apps that provide schedule information. Here's a good Android app: Transport Montreal by Rhatec. It works offline too, and is indispensable when waiting for some of the less frequent routes.
Fares: Buy the 24hr-from-first-use (aka 1-day), or 3-day (expires midnight 3rd day), or CAM Hebdo Weekly (Mon-Sun validity) pass, for easy travel without worrying about transfers or counting tickets.
The bus routes for sightseeing:
15 east: Travels along Ste Catherine St, the main tourist hot-spot in Montreal. Board at the starting point which is the Atwater Metro station, and disembark at the final stop at the Papineau metro.
55 north: Goes along St Laurent, another important street. Take this from Chinatown, near St Laurent Metro, or a bit more south on St Laurent St, and go north in the bus.