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.
165 north: Travels along Côte-des-Neiges, steep road across the mountain. Board at the start, on Rue Guy across from Metro Guy. Disembark in the middle, near the Côte-des-Neiges metro station, and explore that area.
715 east: Goes across de la Commune, in Old Montreal. Take this from Rue Metcalfe St, across from Terminus INFOTOURISTE - the main tourist information office. Disembark at Place Jacques-Cartier, and explore the area.
747 Airport Shuttle: This bus route is nothing interesting but is important to tourists since it travels between the airport and the main interstate bus station (Gare d'autocars de Montréal). This bus is also quite more frequent than #15 on Ste Catherine. So to travel between Guy and Jeanne-Mance (Place des Arts festival area), many times better to walk to René-Lévesque and board this bus. The minimum fare on this bus is the 24hr (1 day) pass or higher.
80 north: Travels on Ave Parc. Board at the start, Place-des-Arts metro and Jeanne-Mance street. Multiple spots to explore: Mont Royal St, St Viateur St (bagels), disembark at Parc station and walk West on Jean-Talon to explore Indian and Pakistani restaurants.
11 west and 97 east : 11 to go into Mont-Royal park, and 97 to travel across Mont-Royal St. Board near the Mont-Royal metro station.
And finally: the Metro itself is a great way to quickly go to a station. Too many to list here - just consult a tourist map which you can pick up for free and get any questions answered at the main Infotouriste office near Peel St cross street Ste Catherine.