Norton 360 is a strong suite which should be able to keep you safe with no problems.What other security products have you used in the past.. ?
You really need to add some on-demand scanners to your config, Tomo172 suggested MBAM and Hitman which are very good , so you should really install them:
Here is a few addons which you can add to Firefox (if you don't already have them)
Additional browser protection:
VTzilla (Free) -
link
VTzilla is a Mozilla Firefox browser plugin that simplifies the process of scanning Internet resources with VirusTotal. It allows you to download files directly with VirusTotal's web application prior to storing them in your PC. Moreover, it will not only scan files, but also URLs.
The scanning options are embedded in Firefox's context menu and download dialog, making the analysis process as easy as clicking a single button.
Adblock Plus (Free) -
link
Adblock is a content-filtering extension for Mozilla Firefox- and Mozilla Application Suite-based web browsers. Adblock allows users to prevent page elements, such as advertisements, from being downloaded and displayed.
LastPass (Free) -
link
LastPass is an online password manager and form filler that makes web browsing easier and more secure.
You can increase your online privacy by using Ghostery :
Ghostery (Free) -
link
Ghostery is your window into the invisible web – tags, web bugs, pixels and beacons that are included on web pages in order to get an idea of your online behavior.
Ghostery tracks the trackers and gives you a roll-call of the ad networks, behavioral data providers, web publishers, and other companies interested in your activity.
You can read
here a review on Ghostery
You should give Sandboxie try, really useful when opening to unknown/suspicious apps or going to shady sites (not that you should do that ..but it happens

)
Watch the below video with Sandboxie at work :
Virtualization:
Sandboxie (Free/Paid) -
link
Sandboxie runs your programs in an isolated space which prevents them from making permanent changes to other programs and data in your computer.