I grew up in the Vancouver area, so I should probably know a few places to suggest visiting if you go there. Some of these are good enough that I still go back when I drive there nowadays.
I tried to go about making this list from the perspective of someone who doesn't want to rent a car for their whole trip, as there is a lot you can do without needing a car or even stepping on a bus.
It may by "just" part of the transit system, but taking the SeaBus across the harbour to North Vancouver is kind of fun. Also the quay (is that what it's called?) where the SeaBus arrives on the north side has some decent touristy shopping and food.
Stanley Park is a must
Can be walked, biked (lots of rental options including bike-cars and tandems)
The route that goes along the sea wall and the circumference of the park is about 10 km and there are trails/paths throughout as well
If you like aquariums, the Vancouver Aquarium is located within Stanley Park
The area that was once the grounds of Expo 86 is nice
Science World is cool for the kids and the Omnimax theatre there is a neat experience
Granville Island is cool for shopping and food
Burnaby Village is another, fancier (than Fort Langley, above) "living museum" focused more on later developments
Craft brews:
Too man local craft beer/etc options to name them all
Along the SkyTrain route:
If you have a car or are willing to take a bus:
Cyprus Bow hiking
Seymour Mountain hiking
Squamish beautiful drive and/or a hike
Lynn Canyon suspension bridge (free city-run park with dirt trails, nice bridge)
Capilano suspension bridge (nicer more accessible bridge but tourist prices and I've never been there)
Fort Langley is a living museum focused on the early days of European settlement and when it was local seat of the government
Skiing is a thing in the winter with four options "in the Vancouver area"