For a very young kid, such as K-2, ChessKid.com is a good place to play. They have built-in measure to prevent anyone from talking to your child without being friends first. The interface is much simpler, and the games are slow so the little kids can have time to think.
For older kids, Chess.com or ICC (chessclub.com) are recommended. That's where everyone goes to play chess, including GMs. You can play different time controls and against players from any countries. Because a lot of adults are playing there, please remind your child that no chat is allowed other than "thank you" and "good game". You may want to monitor the account activities from time to time.