This should be underlined and in bold!! In general it's not difficult to track down the computer belonging to a particular Internet user (sometimes attaching a real name to that user rather than their "screen name" can be a bit of a challenge but don't count on that).
Besides an up-to-date virus scanner, it's essential that you keep up with security updates for your operating system - for example, for Windows systems use the IE option tools->Windows Update. You may not want all of the updates offered but in general you should apply all of the ones that are security-related.
Personally I'm always appalled by the number of people who surf the web with their PC directly connected to the Internet. This exposes all of your computer's network ports to anyone who wants to try to connect to one of them, and if the port happens to be open, all kinds of mischief can happen. It is possible to close the ports from software, or to get a software firewall, but IMHO it's much safer to use a hardware firewall. You can get basic inexpensive boxes that incorporate a simple firewall and a small network switch for under $100 - this would allow 2 or 3 computers in your home to surf the net simultaneously as well as providing an added level of security. The boxes are even cheaper if you get them on eBay - often available under $50.
The Microsoft email apps (Outlook and Outlook Express) are notoriously vulnerable to security hacks in email. My personal recommendation is to avoid them altogether. The two applications I'd recommend would be Eudora and Mozilla 1.6. Both have options for free downloads. Eudora is an extremely well-done email application, and Mozilla is the follow-on to Netscape and includes a web browser, web composer, and email/news application. Both are much more resistant to email viruses than Outlook/Outlook Express, have spam filtering options, and (in the case of Mozilla) popup blocking. However even if you do decide to dump IE, you should still keep it up-to-date since there are some exploits that can use it even if it's not currently visible due to its close relationship to the OS.
The spyware finder apps like Ad-Aware are good to have and a good way to try to maintain some of your privacy on the web, but don't expect that they can provide perfect privacy. The web is like the mall, not like your bedroom. However most of the spyware that these applications remove isn't malicious in the sense of attempting to hijack your computer or financial information; it's "just" trying to record your surfing habits for statistical purposes and perhaps correlate that with who you are in "real" life if it's able to do so.
NEVER, NEVER click on an executable email attachment or download software unless you're very sure who sent it or who produced it. Pictures (*.jpg, *.gif, *.png) are OK - they don't have executable code, or at least if they do, your computer won't try to execute it; but screen savers (*.scr), PIF files (*.PIF), and other executable files (*.exe, *.com, *.bat) can all be very dangerous if they were written by a malicious programmer. This is especially true of files at places like Kazaa!!! (Not that anyone here would use Kazaa :-). Something like 2/3 or more of the programs at swap sites like Kazaa are hacked in some way. If you must download one, put it on an isolated machine that you don't care about :-).
Perhaps it's best to think of the Internet as someplace like a rug bazaar or a flea market where the operative word is always caveat emptor!