There are various ways to really appreciate the beauty of this stunning city. My favourite is the flight in or out of Vancouver. On a clear day, the vistas from your window seat are just staggering. In the city, head for the Convention Centre/Cruise Terminal at the harbour front for great views of Coal Harbour, Stanley Park and Grouse Mountain. Another great way to enjoy the natural beauty of the city is to rent a bicycle (or roller blades) and cycle along the sea wall from False Creek, along English Bay to Stanley Park, then around the entire perimeter of the park (a total distance of about 20 kilometers). This is a very popular route for both locals and tourists alike especially in the summer.
For a total overview of the city and the harbour, head up to Grouse Mountain, a short drive away from downtown Vancouver. In the winter, Grouse Mountain is the closest ski resort to downtown Vancouver.
Downtown Vancouver is very compact, making it easy to visit the city's major sights on foot. One of my favourite spots is Stanley Park, a massive 400 hectare park that encompasses woodlands, beaches, lakes and flower gardens, all connected by a wide network of trails. My favourite route is the sea wall trail that winds around the park - this trail takes you past the marinas (with a panoramic view of Vancouver's skyline and Coal Harbour), Lion's Gate Bridge, the Little Mermaid (yes, there's one here too...just as little as the original!), virgin woodlands, a spectacular rocky coast and the various beaches. The Vancouver Aquarium, a must-visit, is also located within the park.
To get an idea of the history of the city and an insight into the city's architectural heritage, I can recommend a Heritage Walking tour. There are various tours: some routes go through the quaint cobbled streets of Gastown (don't forget to stop by the Steam Clock), Chinatown or the suburb of Shaughnessy (one of the city's more affluent neighbourhoods with stunning heritage mansions).
Another of my favourites is Granville Island. Located just a short hop across False Creek from downtown (on the wonderfully cute and efficient False Creek Ferry), Granville Island boasts a variety of handicraft and art shops, as well as the famous Granville Island Market. Take some time to stroll around this fascinating market and marvel at the freshness of the produce and the huge variety of seafood.