The best way I know that non-technical folk can get as anonymous as possible is to use the free TOR Project, which uses a series of volunteer proxies to route traffic so as to mask its origin.
You can get the TOR Project as a plugin to Firefox and use Firefox in Private Browsing mode to avoid leaving traces on the host computer and you can go even further and use a sandboxed browser, either through running your browser through Sandboxie or in a Virtual Machine, or you can get dedicated sandboxed browsers such as KACE that when run alongside TOR will give you a good chance of staying as anonymous as possible. It is very hard to remain completely anonymous.
Below are the links to all these goodies! :)
- TOR Bundled with Firefox
- TOR Plugin standalone
- KACE Browser