How can I go from PET airport to downtown Montreal?
The simplest choice is taking the 747 bus line. It runs 24 hours a day, 7 days a week and takes about 45 minutes to an hour depending on traffic. It also features WiFi service on lots of the buses and luggage racks. The bus stops at many places between the Lionel-Groulx and Berri-UQAM metro stations.
The ticket costs 10 dollars and includes unlimited access to the STM's bus and metro network for 24 hours.
You can also take a taxi directly from the airport. The taxis assigned to the airport charge a flat 38$ fee + tip and take around 30 minutes to get downtown. Note that if you wish to travel further, the meter charge past downtown will increase your bill.
You can also take the regular bus service. Take the 204 Cardinal EAST line towards Dorval terminus, get off and use the tunnel to go to the 211 Bord-du-lac EAST line towards Lionel-Groulx metro station. From there, simply take the green or orange metro line towards your destination. A ticket costs 3,25$. Keep in mind that the bus does not take bills, only coins.
NOTE: I recommend you only use this method if you are really absolutely strapped for cash and will not use the free 24 hours included with the 747 ticket. It is MUCH less convenient than the 747 line and (in my opinion) not worth the 6,75$ you will save.