The first map posted by Groningen is a bit ambitious: it shows some planned LRT and Subway developments. You'd be better off looking at the
TTC's website for maps of the subway and streetcar sytems.
There are two main subway routes (Bloor-Danforth running East-West, and Yonge-University, running North-South in a 'U'). In addition, there is a grade-separated rapid transit (Scarborough RT) extending the Bloor-Danforth subway to Scarborough, and a "Stubway" along Sheppard Avenue that is symptomatic of Toronto's political paralysis with transit expansion. Think of the endless arguing London's had about expanding Heathrow/Gatwick/other airports, and apply that to TfL.
One of the proposed LRT routes is under construction, the Eglington Crosstown route. In addition to the above, there is a network of bus and streetcar routes operating throughout the inner city.
Virtually all mainline rail is concentrated on Union Station. It's currently at the tail-end of major redevelopment work, to cope with increased passenger numbers. The GO Trains are (provincial) government-funded commuter routes, operated by diesel-hauled double-decker trains. Some routes (e.g. Lakeshore East/West) operate throughout the day, others (e.g. Richmond Hill) only have inward services in the morning, and homeward services in the evening. The newest portion of the GO network is a service operating between Union Station and Pearson airport, which started last year. There are also GO Bus routes that operate further afield.
The local equivalent of Oyster is the PRESTO card. It's accepted by the TTC (subway, streetcar), GO Trains, and surrounding regions' buses.
If you want some reading material before you go, you can while away many an hour on the
Transit Toronto website, which covers the history and current operation of the subway, rail, streetcar and bus operations of Toronto.
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A further post to add: in the downtown area, between the skyscrapers, there is a network of underground walkways known as the
PATH network. These are open to the public, and many have shops along them (or even go through larger stores). If the weather or traffic above ground gets to you, you may be able to walk under it all. A map is available
here.