Using a router you have two networks. Your router's connection to the ISP (WAN), and your computer's connection to the router (LAN). The two are logically separate.
In this instance "Sometimes it says theres limited or no connectivity, sometimes it fails to register anything at all, sometimes it says I have a very strong signal" the issue is your LAN.
In this one, "After going into the router and looking at the system logs, pretty much ever other line says 'authentication error'. (going on click here)", the problem is your WAN. You need to sort out the connection the ISP, and ensure you can access the internet from a PC connected by a cable, before you can tackle wireless problems.
The authentication problem suggests you have the wrong username and password entered into the router.