Now the firefox alert me that is it the old version. However I cannot find the correct update suitable for my Mac. This version is just so slow. Where can I download all the safe stuff for my mac?
The last Firefox version that will work with OS X 10.5.8 is Firefox 16.0.2, but that version is no longer maintained with security updates.
You can consider to install the latest Firefox 10.0.x ESR version that will still receives updates until it is replaced by Firefox 17 ESR (possibly in February 2013).
Firefox 17 require OS X 10.6 or later, so if you want to continue using Firefox beyond Firefox 16 then you need to update your OS.
You can see the Firefox 17 system requirements here: