We now have some things that didn't exist when I was younger: social networking and cellphones. How should a young person stay safe online? IMVHO, it starts with face-to-face dialogue before the computer is powered up or the cellphone turned on (most cellphones have internet access).
--The online names should NOT your own name. Use a nickname or a made-up name.
--NEVER open an e-mail or a link from someone you don't know.
--NEVER give your name or other personal information to anyone online.
--If you must post a picture online, post one that doesn't have anything that would identify your where you live. And no inappropriate pictures.
--If someone you don't know tries to instant-message you online, asks for personal information, or asks for inappropriate pictures/actions, tell a parent or trusted adult.
--Some ISPs have separate sites for children (Yahoo/Yahooligans) or a separate setting (AOL). Take advantage of them.
--If someone wants to take inappropriate pictures/webcam shots of you, refuse and immediately call the adults.
Stuff I've learned over time and reading