Questions like this tend to start arguments around here (because people are loyal to their favourite products).
Let's start with some basic information:
Firstly, if you do decide to ge rid of Norton you can't simply uninstall it from your computer's Control Panel. You need to use a special tool:
https://support.norton.com/sp/en/uk/home/current/solutions/kb20080710133834EN_EndUserProfile_en_us
Next, it's important to remember that Norton provides a complete suite of security programs, not just an anti-virus solution. So if you replace it with a free anti-virus program you might need to consider topping it up with additional security software to perform the other functions that, up until now, Norton has been covering.
The principal free anti-virus programs are Microsoft Security Essentials (MSE), AVG Free Edition and Avast! Up until recently MSE was probably the most widely recommended on here but, after it failed several independent tests, Microsoft have admitted that it's not as good as they'd like it to be.
So that leaves AVG and Avast! Both are exactly the same as you'd get if you paid money for the full security suites offered by those manufacturers, so (contrary to what some people around here would have you believe) they are in no way inferior to commercial products. (They just don't have the 'extras' which the manufacturers hope you'll go on to pay for later).
I used to use AVG but I found that it could slow my computer to a crawl. I now use Avast!, which I'm fully satified with:
http://www.avast.com/index
To add in the extras which Norton has been providing you with, you need to seek an anti-malware solution. The free version of Malwarebytes Anti-Malware should meet your needs:
https://www.malwarebytes.org/free/
Norton also provides its own firewall. While many people are happy to just use the firewall which is built into Windows instead, some people regard it as insufficient and use (for example) Comodo's free firewall:
http://personalfirewall.comodo.com/
I'd always be happy with free security software but, if you do decide to spend money, forget both Norton and MacAfee. They're both bloated and 'resource hungry'. Kaspersky is far better.