Weybridge, like any town, has changed over the years. It's always had an (undeserved ?) snobby reputation. St Georges Hill is a private estate where millionaires and celebrities are rumoured to live.
St Georges College is in Weybridge (it's actually in nearby Addlestone but hey.) It's a private school, it's up to you whether it would meet your childrens needs.
Cobham is a nearby village, which again, is expensive. It's got a large ex-pat American community. Stoke D'Abernon is between Cobham and Leatherhead.
Addlestone,, Weybridge and Cobham have a train station. The trains into London from here are a stopping service i.e. stop at most stations on the way to London Waterloo aka the slow train. Faster train connections to London are from Woking or Guildford. However, driving to either of those places to get the fast train that will travel through your local stations at speed will seem counter productive.
As for community feel/sports. Well, as has been suggested, local authority websites might be a good place to start.
Elmbridge borough council is the local authority for Walton on Thames and Esher. Most other local authorities have the same name.
It's just as well you've got three thousand pounds pcm for accommodation costs. It's going to cost all of that ! Weybridge and the surrounding areas are a real money pit !
Why ? Because it's always been like that.