There is going to be some adult content in the forums, so you may get some questions from your "kid"
My kid plays on chess.com
Any child under the age of 13 should be constantly supervised by a parent or guardian while playing on Chess.com as per the rules of the site. :) It's a good idea to be familiar with the disable chat (for both online and live chess) and blocking features on the site as well.
Keep telling your kid that the internet is not a reflection of how people behave in the real world; it's how they behave when they realize there's no one keeping a list of who's naughty and who's nice.

My (quite young) kid plays on chess.com. I mean the adult part, Chesskid.com didn't interest him.
Is there anything I should keep in mind, do's and don'ts? He is not allowed to chat or upload his picture - but what else?
(So far he likes it and we haven't encountered any problems.)