You didn't mention where the PC is located at. If you are at home and behind a NAT, your IP will be a private/internal IP address and people from the Internet won't see your computer unless you configured it that way (example: port forwarding, etc). So your Internet facing router already acts like a firewall and you have another firewall on your PC setup by default with AV as you claim.

Let's say for the sake of argument you had the PC Internet facing (in your DMZ for example) and you had default firewall settings. If the attacker is on the same network segment as you (which probably wouldn't happen unless they were on your wireless with you on it), they could monitor your plaintext traffic and possibly perform man in the middle attacks on you. By knowing your password, they could run attacks via SMB port (445) to attack your computer. "If" you had RDP open on 3389 they could remote desktop to you but you would have to enable that first. Any applications/services exposing open ports could possibly be vulnerable too.

Making sure you have a computer that isn't Internet facing would be step-one to being safer. This also includes not being on wireless as wireless routers act like hubs in a sense. Also, closing down ports not needed, uninstalling software not used (another example is disabling PowerShell if you don't use it) and hardening your computer by using best practice configurations would help.

What are the "Best" precautions one could take to prevent compromise. Easy, turn off your computer. Seriously though, the above mentioned things are important, having a strong password, updating your software and OS regularly and most importantly, using common sense when browsing the Internet and clicking/opening anything. You can be behind a NAT, have the best AV out there, lock down ports, etc but it doesn't mean jack if you open a malicious file and/or get social engineered. I can write a payload right now that will be custom and not get picked up by any AV. If it never gets to you to run, you'll be safe. The trick is not letting it get on your computer.