We live in South London - the part is called Herne Hill and it's between Brixton and Dulwich. Brixton is the last stop on the Victoria Line and it takes approximately 15 minutes to get to Oxford Circus. Herne Hill does not have a tube station, but has a great overground train that takes you into Victoria in 10 minutes (which is two tube stops away from OC). I really like Herne Hill - it's got a great park nearby some cute restaurants and is close enough to the bustle of Brixton without suffering any of the ill-effects. However since you have children, I would suggest taking the plunge into Dulwich, which is similar to Herne Hill, a little further from Brixton, but more residential.
There tends to be a huge divide between North of and South of the River here - my experience is South.
Good websites to check out are findaproperty.com, upmystreet.com and gumtree.com (good for private rentals/sales)