Be aware that you can only obscure your real IP address, not completely hide it. If your computer didn't transmit your IP address, you would not be able to receive any responses back from a website (your IP address tells the web server where to send the web page you are viewing).
You can use certain techniques to make it more difficult for you to be tracked down, but if someone (say, the government or a hacker) wanted to track you down, they could. The average webmaster, however, may not know the difference or know what to do to track you down.
And hiding the referrer really is useless and will actually break certain sites (for example, if a website requires you to visit one page before viewing another, it uses the referrer to tell if you visited one page before the other page.) Also, all the referrer does is say what webpage you came from if and only if you clicked a link on that website. The referrer actually tells nothing about you or your IP address at all. Just what the last web page you visited was if you clicked on a link.
The closest thing to anonymous surfing is using a service that acts as a middle-man and gives their IP address instead of yours. And I am not sure how trustworthy all of them are. Some may even be set-up by the government.
So, in summary, there is no foolproof way to become untrackable. So be careful.