Stanley park is nice for kids. There is play equipment, beaches, and a bike path/walking path that circles the park (you can rent bikes, double bikes, possibly trailers) and runs alongside the sea front (with a seawall). Stanley park also has an aquarium.
Weather is very nice in August, usually summers are not too hot. It does rain a lot in Vancouver and unfortunately usually when it does rain it rains for AGES (if it's raining in the morning it will often rain all day!)
I don't advise wandering through the east part of Vancouver, very awful (sad) even during the day, East Hastings street particularly.
The sea bus (public transportation, linked to the skytrain network) is a cheap way to cross the harbour and will take you to the north shore and the Lonsdale Quay Market.
Grouse mountain (on the north shore) has a nice view and operates the tram (called the skyride) during the summer.
Capilano Suspension bridge (also on the north shore), good fun, maybe a bit scary if you're the nervous type (I am!).