Everyone has given you some great ideas - Granville Island, Science World, Stanley Park, the Aquarium... if you are staying downtown, try taking the Aquabus if you get a chance - they link from "downtown" to Granville Island and Science World. It's really fun for kids - whenever my niece & nephew visit, they want to take the Aquabus over to Granville Island (we live in downtown Vancouver). If you decide you want to take a picnic into Stanley Park (it's a great place to picnic in the city), there's a local gourmet grocery store called Urban Fare, that has picnic type food ready to go. They're not a cheap deli, but the food is always good and they'll have some kid-friendly options.
Another fun thing to do is rent bikes and bike around the Seawall in Stanley Park (the whole loop is around 10k, but there's trails through the park that can cut that down if you don't want to do the whole thing). There's MANY places to rent bikes along Denman St (near the entrance to Stanley Park, you can google map it

). Most bike rental places will also give you maps and such as well.
One thing I would point out though, is if you decide to head out to the Museum of Anthropology at UBC, pay attention to your time frame if you're going to take public transportation. It can be CRAZY to try and get a bus to UBC (from downtown) during the times of 8-10am and from UBC from about 4-6pm during September. The busses are frequent during this time, but also VERY packed (I used to do this everyday

), so if you don't want to be crammed into a bus, try the off peak hours.
If you take the train up, the train drops you off in a not so nice area of town, so try to schedule your trip to drop you off before dark if possible. If you take a cruise ship up/down, the cruise ship will most likely drop you off at Canada Place (in the downtown core), which is a nice area.
As for accommodations, I'd also recommend the Priceline option. If you can, uncheck the option for the Granville St. area when you're searching - it's the "entertainment" district and has a lot of clubs and bars, so with a family, you're probably not going to want to stay there.

The West End, Coal Harbour, Yaletown areas are all nice areas if you see those options and all are very family friendly. I've had family stay in the Coast Hotel, the Sylvia Hotel, Pan Pacific and the Westin Bayshore and they all felt like they were family friendly.
If you're going to take public transit, the Translink website (I don't think I can post outside links, so you'll have to google it) will give you schedules and route information and have a handy feature to map out your route. If you have a question about some of the routes, let me know - I'm a frequent rider of the transit system here
