Watch out for the dreaded grammar systems.
State schools in some areas still have grammar schools. To get in you child must score highly in the 11 plus test, sat in Year 6 (10 -11). The highest scores get grammar school places, all else go to the other state schools. Little chance of your kids getting into grammar schools.
Kent, Medway, London boroughs Sutton and Bexley are all grammar. There are others.
Surrey is the comprehensive system. East Surrey has decent trains to London Bridge and Victoria, and is cheaper than further West, which tends to go up to Waterloo. You can walk over London Bridge and be in the City in 5 mins, it's a nice walk.
Look at Whyteleafe/Warlingham/Sanderstead, it's Zone 6 (so still London for train ticket purposes). Warlingham School and nearby Riddlesdown are excellent schools. De Stafford in Caterham is smaller and less good, but OK.
Further out, Merstham has stupendous links to London Bridge (but some quite hairy bits) and is outside the Oyster zone for trains, so a pricier commute. St Bedes is a great school (ecumenical RC and C of E).