In the city itself, do not miss Stanley Park, which has wonderful drives plus hiking and biking, spectacular views of the city, and amazing killer whale and beluga whale exhibits. In terms of the killer whales they have shows, but the whales don’t do tricks, they only have them demonstrate natural whale behaviors such as how they stun the fish they eat. Very worth seeing.
Vancouver is a large city with a very European feel. It has an excellent restaurant scene and wonderful shopping. Granville Island is one good choice for both of those.
The Sun Yat Sen Chinese Garden is wonderful. Very calming after hectic tourist hours or days.
My very favorite cheese in the world is Black Diamond Aged Canadian Cheddar (at least two years old). It is so sharp, it almost bites back. Try some while you are there and/or bring some home.
I second what @gailcalled says about Whistler. The drive there is spectacular. The Village is pretty commercial, but the outdoors scenery is exquisite. More expensive than renting a car, but a fun way to go is the train out/flight back day trip described here.
It is also easy to take a ferry from the Vancouver area to Victoria, B.C., which makes for a fun day trip. The two cities have very different characters. Victoria has a bery British feel rather than Vancouver’s continental flair.
Have a wonderful time.