Should I worry about my kids using the Internet?
No. The stories you hear about children getting in trouble on the Internet are rare and highly exaggerated. In fact, if you look closely at the warnings, you will see they usually come from adults who are scared of life and are trying to scare everyone else. Every day, millions and millions of kids around the world use the Internet in perfect safety.
However, your kids, especially the youngest ones, will do better on the Net if you help them and guide them. You do need to teach them (1) how to protect their privacy; and (2) how to make active judgments about the quality of the information they find on the Net.
My suggestion is to use the Internet with your kids. As they get older, you can let them use the Net more and more on their own. But make sure they know that, as the parent, you always have the right to look at what they are doing, including their email.