If you know Vancouver, please contribute your ideas
From the Airport to the Hotel
The hotel is the Westin Bayshore, which is located at 1601 Bayshore Drive, Vancouver, BA, V6G 2V4. It's located in the North West of the city centre, near the start of Stanley Park.
The SkyTrain Canada Line light rail runs from the Airport to the Waterfront Station in downtown. In peaks times, trains run every 6-7 minutes, and the journey takes just under 30 minutes.
From the Waterfront Station to the hotel is about 2km, so you probably don't want to walk (unless you don't have a lot of luggage!). One option is to hop in a taxi for the short ride - there should be lots of taxis near the station, and the fare should be around $10. Alternately, get off at the City Centre Station (one stop before), which is on West George Street. Catch a 240, 250 or 980 bus heading west, which stops on West George Street almost outside the hotel. See the TripPlanner for details.
If catching the SkyTrain from the station, and you buy your ticket at the airport station, there is a $5 "YVR AddFare" surcharge added, taking the trip cost up to $8.75. This only applied to tickets bought at the airport station, so you can avoid it (and save the $5!) by simply buying a $3.75 FareSaver ticket at the PharmaSave or 7-Eleven located within the airport.
Alternately, a taxi from the Airport to the Hotel should be around the $45 mark.