There are several different buses and shuttles that run to and from the airport around the clock to different points in the city. To get downtown, expect travel time to be about 45 minutes. A taxi will get you there for a flat rate of around $40.
The Toronto Transit Commission (TTC) is undertaking a massive project to update and extend the streetcar system that will be known as the Light Rail Transit (LRT) to allow better access to the north, west and eastern parts of Toronto. Regular fare is $3 and the current U-shaped subway line will generally get you within walking distance of anywhere centrally located. Maps and schedules are available online.