Changing your IP cannot really be done by the browser itself, and the only ways I know how to do this is via contact with the ISP, or a router reset if you know that you get your IP from a pool of IPs.
This means, the browser is using some type of dynamic proxying layer--something like The Onion Router. As they advertise it to be, don't rely on it for pure anonymity. (However, it did thwart the FBI trying to take down a child porn site, so I say it's good enough.) I recommend downloading the TOR browser bundle, which is Firefox accompanied with the TOR router.
There are a couple addons that work with the browser bundle, that provide extra security:
Ghostery--an anti-tracker/anti-cookie addon, and Ad Block Plus (And it's accompanying addons) with the filters `EasyList', `Fanboy's Adblock List', and `EasyPrivacy', along with other lists that fit in with the languages of the sites you're visiting.
Lastly, if you really are the Paranoid Parrot, I recommend getting a VPN and look into I2P.
I also recommend getting into computer security to get a better understanding of privacy, therefore boosting it.