I've had one or two interesting experiences helping peeps who have Norton 360 installed, but to tell the truth they seem to have been one-off events, and you can get the odd misbehavior from almost any program.
I'm somewhat anti-Norton, simply because I like to be more involved in my computer security, and Norton can be a bit of a resource hog, but what I'd say is: If it has been working well for you, keeping things secure, you are happy with it, and don't mind paying the renewal fee, keep it.
If you feel like a change of view, there are quite a few very good alternatives. (Including the one I use, Avast, which, simply, just works.)
Whatever you choose, try before you buy. All decent programs should offer a full fledged trial period of 15 - 60 days; irrelevant for the free versions.
Whatever you choose, download the full installer file for it, and any removal tools you might need before disconnecting from the net and uninstalling Norton (or any other AV.) Here's a link to the
Norton Removal Tool (which is updated as required by MajorGeeks),
Here is a site with links to most other uninstallers one might ever require.
The normal procedure is to remove a program via the control panel, reboot, run the uninstaller, reboot, install the new program, then reboot (usually.)
Don't forget if removing Norton360 and installing one of the lesser featured AV's, to make sure the Windows firewall (or an alternative) is turned on before reconnecting.