Pretty much everywhere in and around London is close (walking distance, even with toddlers!) to shops/parks/schools.
I think a good start for you would be to check out what is within budget on http://www.rightmove.co.uk/ - you can input an area, such as Regent's Park, then increase the radius within which to search until something comes within your price range (you can specify budget too, of course). The maps then show you how close the primary and secondary schools are.
The Tube Map http://www.tfl.gov.uk/assets/downloads/standard-tube-map.pdf will help you see where's where, though it can look a bit misleading, e.g. Wimbledon and South Wimbledon look miles apart on the map, but in reality, they are not very far from one another.
There are also trains that go to Victoria, that aren't represented on the Tube Map, you could look that up and see what a reasonable commute would be, then look there.
For commuting to Victoria, I'd look around Wimbledon, Richmond, and expand your radius until you get a price you are happier with. Wimbledon and Richmond aren't cheap, though they are very lovely, but areas around them, such as Colliers Wood, Tooting, Southfields, are cheaper, and close to Central London, as well as being close to the bigger towns of Richmond or Wimbledon. If you are wanting to be a bit further out, Barnes is fabulous, really gorgeous, on the river, or Putney, or Fulham. Further yet, typical commuter belt, would be Windsor, Egham, Virginia Water, Chertsey, though anything in the South West is on the pricier side of things. Ashford (Middlesex, not Ashford Kent) and Staines might be a bit cheaper. Staines is on the river, but for some reason doesn't have quite the allure of Richmond or Barnes. Clapham is great too, it has a common (big green fieldy area) and is a main hub for the trains, so good to get out and about.
Chiswick is truly wonderful, I would have loved to have lived there. Ealing is nice.
I don't think you can judge a place online though, I'd have to go and get a feel for the place.