Your best bet is to get a computer capable of running the latest version of Mac OS X - Mac OS X Mountain Lion (10.8). This is the most secure option and will allow you to run the latest and greatest Firefox.
That is a pretty crazy suggestion just to update a browser! There is another alternative if you are worried about having an unsupported browser.
You can run TenFourFox whose purpose is to support older Macs.
more info about it here;