One of most popular pages on about-the-web.com is about avoiding scams, hoaxes and urban legends on Internet (http://about-the-web.com/shtml/scams.shtml). Here are a few ways to avoid some of perils associated with being connected to rest of world.
As a general rule of thumb, beware of any offer that sounds too good to be true. This applies to products being offered on Internet and especially to any unsolicited offer or spam you receive in your email. Many of these are scams.
The term "spam", in case you don't know, refers to any unsolicited bulk email. If it's not addressed to you, or you don't recognize sender, or you're being asked to buy something or pass along questionable information, then that's spam. My advise is to "can spam" by immediately deleting any email you receive from anyone you don't know.
The best way to avoid spam, is to be very careful about who you give your email address to. My advise here is to set up a free email account and to give this email address to anyone you don't absolutely trust with your personal information. It almost never does any good to fight back against spam. Your best course of action is just to delete any unwanted email.
A trick that has been pretty successful for me is to set up an email rule (or filter) that automatically sends any mail not addressed to me to a special folder. Most of this is spam and can be easily disposed of.
In addition to spams and scams, other thing to watch out for on Internet is hoaxes, urban legends and false information. There are many of these floating around Internet these days. Beware of any email that asks you to send money for any cause, or to forward email to all your friends. Almost all of these are hoaxes. An excellent resource for information on hoaxes, urban legends and false information is http://urbanlegends.about.com/index.htm