Chessington World of Adventures is one of the most family friendly parks in the UK, is just outside London, and its really easy to get to by public transport:
First you have to get to one of these train stations: Waterloo, Clapham Junction and Wimbledon. From there, take a train to Chessington South. You can get train times from http://www.nationalrail.co.uk/
Once you leave Chessington South station, just follow the signs to the park, its a 15-minute walk.