The best way to get downtown is to get a cab, which is approximately $26. Alternatively you can catch the 'Airporter' coach for around $12 which will take you directly downtown or to the ferry terminal to Vancouver Island.
Updates
The cheapest way, and it's pretty convenient is to get the regular public bus. There is a shuttle bus in the airport that leaves you at the stop, then its about a 40 minute journey to downtown Vancouver. The bus stops right near Robson st (the main shopping street in downtown) and also by the Skytrain.
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