The official build of Fedora Core 6 doesn't include Firefox version 2, rather the outdated 1.5. Fortunately, a French RPM repository seems to fill the gap.
Apparently, you can also get Firefox 2 from the development branch of Fedora - simply run
yum -y --enablerepo=development update firefox
and it will install the fox.
To elaborate, this entry in the Fedora Project Wiki briefly explains, why there is no official Firefox 2 RPM build for Fedora 6 and gives a few other alternatives.
Now, before you grab yourself the latest and the greatest Firefox, check that all your favourite extensions, those without which you cannot live, have been successfully ported to Firefox 2. If that's not the case, you can try installing the
distributive from mozilla.org - but run it as another user or create yourself a new profile, otherwise it will override your current profile and cause problems with the previous version of the browser.
You should also consider the beta of the Flash player for Linux - many newer sites don't function properly with Flash 7, unfortunately.