But if, for any reason, you really need to be stealthy and private while surfing the web – use Tor network to completely hide your identity. There were times when you had to do some dirty work to set up Tor on your computer, nowadays just download the browser bundle which includes preconfigured standalone Firefox browser for any OS. Just start the bundle and use any online service (e.g. this one) to check your IP and other system information. See, they are not yours anymore.
The downside is the significant loss of connection speed and no usual plug-ins like flash for example (you are free to install them, but they can reveal your true IP address). As an extra feature you can explore darker and creepier parts of the web, known as Invisible Web or Deep Net – the .onion domain. Here is a search engine to start with, be advised though – at times these parts are really creepy.