Best Family Neighborhoods in West London?

Hi everyone.  I am accepting a role in Richmond and we are moving permanently to the UK.  My husband is British so this is great for our family.  We will have two small children, an infant and a 3yr old as well as a labrador.  We'd love to be walkable to a good park and close to a District line tube station as my husband will work in Central London.  Any opinions on great areas to live?  We could spend up to 3000pcm but want a 4 bedroom house.  Thanks!!! Jennifer

Richmond, obviously, but Kew, Sheen, St Margaret's or some parts of Twickenham are all close enough to  Richmond and the District line.  Great for families, lots of green space and parks. 

I am not sure about rent, but I think that £3000 a month should be adequate even in that area for you to find a nice place.

Chiswick, Barnes and Putney as well (though Kew or Richmond would be my first choice).