There are a number of walks and hikes in and around Vancouver.
From a very basic perspective, Vancouver is a very walkable city, particularly around the downtown core. Vancouver has a pretty extensive seawall which take you through some very picturesque parts of the city. We take walks along it regularly. Some parts might be better served riding a bike like around Stanley Park.
There are also extensive trails around the city and into the north short mountains with varying degrees of difficulty. This is one of the websites we use for ideas and reference. Note, some of the trails are no joke with unprepared hikers sometimes needing to be airlifted out by local search and rescue.
One of the popular well know trails is the Grouse Grind which is a trail which leads up to the Grouse Mountain resort area. You typically hike up, have a beer, and take the tram down.
Apart from the walks and trails, Lynn Canyon is nice. We like to visit Granville Island (along the seawall). Daytrips to Squamish/Whistler, Vancouver Island, Gulf Islands are kind of nice.
In the summer, there are a number of festivals and events around town that are pretty fun.