It depends how far you want to go really.
In Vancouver, definitely check out Stanley park. Check the totems and lions gate bridge. Also the aquarium if that's your thing and there's a nice restaurant in the style of a hunting lodge close by (name escapes me).
Gas town is a bit touristy but worth checking out, some good food and good drinks to be found there.
Check out the beaches (again the names are gone). I think second beach and jericho were my favourite. Sunset beach on the English bay at, surprisingly, sunset is beautiful and within walking distance of restaurants etc in the evening.
Granville island is pretty cool, check out the brewery!
On the North shore, there's some nice walks around the suspension bridges. Lynn Canyon was free!
I didn't go to the University, but was told it was also worth checking out.
Outside of Vancouver, you can either take the ferry to Vancouver Island. Victoria is nice, probably only worth a day trip though. We went on a zodiak tour (whale watching). Had an excellent time, got thoroughly soaked but seeing the whales in the georgia stait was incredible. Also on the island if you're staying longer I would drive the coast, sooke basin or there's the surf further up the west coast, may be the wrong season.
If you're heading East, we stayed in the Okanagon. Lovely place. Interesting fact, it's actually a desert (or so I read). The climate is amazing, really really hot but then there's the lakes, try wakeboarding, or just laze on the beaches. Check the kettle valley railway if you like cycling and definitely find time for some wineries. I went to Mission Hills. Tour was a bit pretentious but enjoyed the setting, lunch and the wine.
If you've got the time, maybe take the CPR (train) to Banff. I love the rockies. The lakes and the walking are the obvious things to do. Banff is nice for a day, check out Bow falls (near the Fairmont), the hot springs (cave and basin is under restoration) and maybe consider walking up Mt Sulphur. Around Banff, Lake Louise et all are simply incredible. Some need you to sign up way in advance to see them so start checking now!
Either way, Canada is amazing, you'll have an incredible time.