Questions like this one always seem to get people arguing!
Let's start with the straightforward bit. Before installing any alternative anti-virus solution, you need to uninstall Norton. That can't be done directly from your Control Panel. You need to download and run a special program:
https://support.norton.com/sp/en/uk/home/current/solutions/kb20080710133834EN_EndUserProfile_en_us
Up until recently, Microsoft Security Essentials was probably the most widely recommended anti-virus program here (largely because ChuckFickens, who works in IT, gave it his backing). However Microsoft have recently admitted that it's not as good as they'd like it to be, so we can probably rule it out from consideration.
I used AVG (which is also popular) for many years but I gave up on it when it seemed to be slowing my computers down. My current preferred program is the free version of Avast!, which I'm happy to recommend:
http://www.avast.com/index