There are detailed instructions at Symantec's site on how to completely uninstall Norton's AV, but it's complicated.
First, uninstall Nortons from Control Panel/Add Remove programs, or Norton's own uninstall facility.
Then, under Control Panel/Folder options/View/Hidden Files & Folders, check 'Show all files and folders' and uncheck 'Hide extensions for known file types' this will enable you to see all files.
Then search around everywhere and delete everything Nortons/Symantec related. Program Files and Application Data folders are good places to look.
Download
Reg Cleaner, it's an old freebie registry editor, and use that to clean out Nortons/Symantec from registry.
After all that you may find there's one or two files that won't budge. To get shot of them, boot into safe mode, rename the file extension(s), reboot, then delete them.
That worked for me, a tedious process, but well worth it.
I use Antivir, it's free, I'm happy with it, but I do hear nothing but good things about NOD32. Good luck.