Chrissie, I'm sorry that this has remained unanswered for so long.
As you say, the idea is to download the new version and save it somewhere handy ( such as the desktop) , then uninstall the old and then install the new.
For uninstalling the old, it may also be a good idea to download the Norton uninstall and clean up tool and run it......you may be asked to reboot at that point.
Then install the new.again you may be asked to reboot at that point.
I don't think you need to disconnect from the great interwebby when doing all that...in fact the installation may require a connection as some of the progs. download only an installer and require to "phone home" for the rest of the files.
Is your firewall part of the Norton suite, or are you using another one.......if another one that will protect you.