I've never used the Firefox installer, so I can't tell you how to make it work. I am familier with the tarballs so I'll describe that. Get a copy of Firefox
then untar it in your home folder. You could right click on it and choose extract here or you could use an application like file-roller. If you want the command then this will do the job
tar xzf firefox-i686-linux-gtk2+xft.tar.gz
then you run the script in the folder called firefox. I find it convenient to make a launcher on my desktop. Right click on your desktop and choose Create Launcher (that's if you're using Gnome - I don't know KDE but I imagine it's similar). Call it what you want but for command fill in the full path to the firefox script. If you untared in your home it will be /home/$USERNAME/firefox/firefox or somesuch, /home/parttimenerd/firefox/firefox?