If the downtown areas do not hold any interest for you, consider a trip to Grouse Mountain on the North Shore.
Take the challenge to hike to the top via the Grouse Grind (2.5km with 800m elevation!) or ride the gondola ($82/adult).
Spectacular views await at the top, plus rescued grizzly bears and a lumberjack show. Hikers can pay a discounted fee to get the cable car back down ($20 when purchased online in advance).
Grouse Mountain operates a free shuttle bus from downtown Vancouver during the summer months (full ticket holders only, not Grouse Grind hikers). Otherwise, take the SeaBus ferry from downtown Vancouver to North Vancouver and then catch the #236 bus.