The Capilano Suspension Bridge is a huge tourist trap. It is very expensive (30$) and very crowded. No matter how many photos you see of people standing alone on the bridge in advertisements, YOU will always be sharing it with a hundred other people. The Lynn Valley Suspension Bridge, however, is the same idea (skinny bridge over deep canyon), cheaper (free) and is never crowded.
Whichever you choose, both are accessible by public transit. We have an excellent transit system with most touristy areas accessible by train from the airport and nearly everything else accessible by bus or sea bus (including the nice restaurant at the top of Grouse Mountain).
The Museum of Anthropology is completely worth the trip. I can’t recommend anything else any higher.
Stanley Park is a must see. It’s rare to see that much green space smack in the middle of the city. It’s filled with its own amenities (the Vancouver Aquarium, HMCS Discovery, Prospect Point, et cetera ad nauseum) and its own little free transit trolley complete with tour guide (he accepts tips, if you choose).
That’ll also be post-Olympics, so it might be fun to tour some venues. GM Place (ice hockey) holds tours twice a week that take you through the whole building (dressing rooms, broadcast, fancy suites and such). BC Place also runs tours, though I’m not aware of their schedule. The BC Sports Hall of Fame is also there.
Granville Island is an old industrial park converted into a public market and art district. It’s a great place to shop for very unique gifts, fresh produce, art and beer (the Granville Island Brewery also holds tours and tastings). There are always street performers during the summer. It’s just a fabulous atmosphere.
Restaurants are hard. Everyone has their favourites and Vancouver has a huge, crazy variety. The best (and most unique) hot dogs are at Japadog. The only street vendor that ever has a lineup circling the block. My favourite burgers come from Splitz on Main St or Burger Heaven at the New Westminster skytrain station. Asian food… well… I can’t even pin it down. Whatever you try, it’ll probably be at least all right, and possibly be incredible.
That’s off the top of my head. I love my town. 