At least they told you they would no longer be providing support, they never told me until my computer got mucked up.
Anyway, that aside, Norton can be a pig to remove. Start by trying add/remove programs, then you can try ccleaner for any bits left behind. If that doesn't work there are removal tools on the Symantec website and someone could supply the link if required. But Symantec only recommend using the removal tools after trying the add/remove programs method and it doesn't work. I presume it works is some cases but not all.
Download whatever anti virus you want to the desktop before you remove Norton but don't install it until you get rid of Norton. I like AVG myself.