You cannot be anonymous on the internet
You can get some of the way there for everyday use. Tunneling encrypted proxy service such as Guardster, Proxify, Shadowsurf, Anonymization, Anonymizer, Tor, Snoopblock, etc. Work with a service in a nominally secure country (i.e. not most of the US ones listed above). Sweden's got a couple. As long as the session is encrypted to the proxy, you're 90% done since most logging at the superficial level is done at your local ISP or maybe NIX.
Anyway, hiding your IP address only hides it from the web site logs, etc. Cookies and beacons will still follow you around unless you use a service that strips them out.