The internet is an invaluable source of information. You can learn a great deal about how it works by surfing sites related to internet safety. Don't forget that there are many people capable of many things, and if you don't know about them, you won't be completely safe. Ignorance is not always bliss. Use common sense.
Common sense is generally a good guide in the chat situation also.
Do not forget that people CAN be quite different than they present themselves as being. While there are wonderful, intelligent people, there are also those who are not. Just because you're ok, doesn't mean that they are. Be careful, and trust your intuition. Share with them the dangers of giving out information.