I would be surprised if you can get rid of Norton completely, but one firewall is enough and two (even the XP built in one conflicts) so stick with zone alarm only and it will do all you want it to do - have used it for 5 years and it has not let any nasty through yet.
Go into msconfig and make sure you do not have any residue of Norton running at start up, and disconnect the built in firewall (if you are on XP) and you will have no problems - apart from a slower start up that ZA gives, but that's no problem.