Vancouver. I am British and moved here in February. I love this city! You're right, it will be cold. And it is very wet. A rain jacket or umbrella are very useful. Firstly, stuff you MUST know about the transit here - it's breathtakingly efficient. The bus drivers don't give change, so you'll need 1.75 in loose change for each journey, or buy a book of 10 tickets in advance at a 7-11 store. When you pay on the bus, get a "transfer" from the driver and this entitles you to unlimited travel on the buses, seabus, and skytrain for two hours. The zonal system can be confusing (applies during weekdays). To get from the airport to downtown, take a number 100 to 70th Ave, then a number 8, which says 'Granville' on the front. Since this is a two zone journey, you'll need 2.50. Every coin has a nickname; 1c - a penny, 5c - a nickel, 10c - a dime, 25c - a quarter, $1 - a looney, $2 - a tooney. Beware that the prices displayed in stores are not what you actually pay. 14% Sales Tax is always added.
The Jericho Beach Youth Hostel is great, and the view of the city from the beach is spectacular. You should see Lynn Canyon on the North Shore. Most people are desperate to get to the very hyped Capilano Suspension Bridge, but Lynn Canyon is better and free. The Harbour Centre Tower gives a good view and costs $9. Your ticket allows you unlimited access for the day, so you can go up after dark as well as daytime, or even watch the sunset there. Walk down Robson Street in the evening - it's full of light and colour and a great view of North American glamour shopping. If you feel culturally inclined, go to the impressive UBC museum of Anthropology. One of the best restaurants is Steppho's Greek Taverna on Davey Street (near the junction with Thurlow Street), where there's huge portions for $9. There are also queues around the block if you get there after 5.30, so be very early!
Right now is the ice hockey season, and I definitely think you should see a game! Finally, if you have time, I recommend a trip to Victoria and/or Tofino. At Tofino you can watch whales or walk in the rainforest.