Stanley park sits right off of English Bay beach and provides a great little nature oasis in the heart of Vancouver. There's no shortage of trails to explore and you can bike/walk/run the seawall. There are a number of beaches along the way to visit and during the summer a ton of events happen either at the park or right next to it.
Stanley Park has everything from pitch-and-putt to an aquarium, Brockton Oval where you can see cricket or rugby games, and TONS of beautiful green space for walking, picnics, flying kites, and spectacular views of the oceans, mountains, and the city.
Bypass English Bay in the summer and walk the seawall to Third Beach - English Bay is always packed in the summer, third is where Vancouverites go.