It's not that nice a city anymore. It's overcrowded and there are homeless people sleeping on the downtown sidewalks.
But I guess the most interesting neighborhoods to check out would probably be Gastown, the West End, Yaletown and Kitsilano/Point Grey. White Rock/Crescent Beach are nice if you want to drive out of the city. I grew up in White Rock. It's right on the US border. That would be to the south.
To the north across the inlet is the North Shore. Ambleside and Horseshoe Bay are places worth visiting in West Vancouver. I live in Horseshoe Bay village myself.