Fester:
Anonymity on the internet is tricky to achieve. You leave a trail whenever you visit any sites and through that trail it's possible (although difficult) for the unscrupulous to track down your own computer IP and from that - your real name and address. All the systems which try to achieve anonymity slow down your browsing, too, so there's a compromise in terms of speed and safety - just like driving, in a sense.
The most effective free solution is Tor. Once you download it, it patches into Firefox directly and then you simply run the program. There's no techie messing about. Your browsing is then routed through about 100+ servers around the world to the one you're actually looking for, so if anyone intercepts your browsing they'll only see the servers through which the information is being bounced.
It's not 100% anonymous: no system is, but it's as near as you're going to get for free.