"Kid" is very informal. You'll want to use "child" unless you are speaking with friends. ("Kid" isn't rude, it is just informal.)
"Baby" is more conversational than "infant". For instance, if you are talking to someone, it is more natural to say "What a lovely baby" than "What a lovely infant". One example of when to use "infant" is when describing things, such as when shopping: "Where can I find infant clothes."