I only joined this forum today so I may be too late with my advice. Hopefully your own research has yielded good advice for your first-time Canada visit.
Anyone who hasn't been to Vancouver or Canada MUST go visit Stanley Park, a large public park which is a peninsula jutting out on the northwest part of the City. You can drive counter-clockwise around the park and make frequent stops in the parking lots located at various scenic and strategic places (restrooms, food, viewing spots). These parking spots used to be all FREE but as in all places now, FREE parking is a fantasy. It's not that expensive, though.. pay one fee at the dispenser and you get several hours at all the parking spots in the park. While driving, you will generally have to your left, the park area with the tall oldgrowth trees and to the right, you'll have the water. I would skip the zoo and the aquarium -- Americans have better zoos, etc.
Another must-see spot is Granville Island Public Market -- a foodie/cook's haven for fresh produce, fish and other foods you can take away to eat or cook. I'm not that impressed with the restaurants or the little food kiosks (not bad, but nothing to write home about). Peruse the food offerings in the market, get a takeaway... if the weather is dry, take it outside and find a seat in front of the water. Watch out for the seagulls, though... or else eat inside in the seating area and then take a walk outside the market. There actually is free parking for about three hours before you get ticketed, but these go quickly. I would park in the paid parking area -- just drive around (generally one way to drive once you're in the "island") and you'll see where the parking -- free or private-- areas are).
Science World is also a great place for parents with kids. Exhibits are always changing and there are places where kids can actually play and touch the stuff. There is parking (paid parking via dispenser) outside the facility. You can spend several hours there. Outside Science World, there is a walkway, go left (facing the water) and you go towards Creekside Community Centre (there are washrooms there)... keep walking and you'll have the water to the right and you'll see downtown across the water. Go right for a bit (about I'd say 20 minutes) and you'll end up downtown (or the tip.. not in the middle of downtown)... stop by "Urban Fare"... kinda like a smaller Whole Foods with a cafe and food counter in the front for a snack and washroom break before heading back.
The centre of downtown Vancouver is Granville and Georgia -- as in any city, parking downtown is horrendous. The way to get there and just wander is to take public transit -- our SkyTrain system is pretty good and frequent. Bus service is pretty good... better depending on which bus... info is available on the net. Walk towards the actual centre of the Financial district, a few blocks away around Burrard Street and West Pender.
I would skip Chinatown and Gastown, but that's a personal preference. If you want good Chinese food, I wouldn't look for it in Vancouver. There is a suburb just south of Vancouver called City of Richmond which pretty well has all the decent Chinese food. If you have a base you are staying in Vancouver, I would visit Richmond via the "Canada Line" SkyTrain going towards "Bridgeport" (not the airport). Stop at Aberdeen Centre Mall stop and go to the top level and get to the food court for a snack -- mostly Asian stuff... or if you are more serious about your Asian food... near Aberdeen Mall, go to the next mall over "Parker Place" for more food court Chinese food or find Alexandra Road (a few more blocks over) for the Chinese version of "restaurant row". In any case, if you are around this area, there's no end to Chinese restaurants... in general, the high volume of restaurants ensures good quality and good prices. I have a favourite... not on Alexandra Road but nestled in a small outside mall -- "Empire Mall".... it's a busy, high turnover restaurant frequented generally only by Asians -- "Double Double restaurant"... order their congee -- rice porridge, fish chow mein and/or yung chow fried rice. If you know your Chinese food, order their fresh made chinese doughnut to eat with your congee.
If you have time, I'd take a daytrip to Victoria...wouldn't actually stay over. The 90 minute ferry trip and the view is really the thing. The City of Victoria itself is pretty but it's a government city and most kids would find it boring... I'd drive to the Tsswassen ferry terminal (south of Vancouver) , park it and do a walk on on the ferry -- bringing a car into the ferry costs way more money... enjoy the 90 minute ferry ride... once you land in Schwarz Bay ferry terminal, you take the #70 bus right there outside the ferry station to the middle of Victoria and back. The bus ride to Victoria is a bit slow... about 45 minutes if I remember correctly... be patient. Check on the net for ferry crossing times.