If this Norton is an all in one with virus checker and firewall then you need to ensure that you install a new virus checker as well.
As I understand it you have tried unsuccessfully to install Zone Alarm and you still have Norton installed
If this is right then you need to uninstall any Zone Alarm files that have installed. Try using add/remove programmes or a zone alarm istaller if one has arrived in your list of programmes. Otherwise search for Zone Alarm files via start>search and remove them and any folders manually
Then you need to download a new virus checker. If you get (free) Avast you can download the programme and the up to date database as two separate downloads. Once you have done that, disconnect from the internet, uninstall Norton, re-boot and open the Avast installer. The new virus programme will install and after that you can click on the database and the up to date virus definitions will install. Re-boot and then you can install Zone Alarm from the disk or whatever.
As Fruit Bat says, Norton can be quite difficult to uninstall. They do have a tool you can download