Alright,
So when I'm trying to download Gwent using GoG Galaxy it pops up with a screen that says "Your code" and it keeps asking for a code. I never got one of these codes and if I did I have no idea where or how to find it. It won't let me install Gwent because it says that I'm "Not authorized"!

I have absolutely no idea how to fix this and would really like to play Gwent. I thought this was a PUBLIC beta!

If we could get this resolved quick that would be appreciated! :D

The code is 'bacon' by the way.

That looks to me like you try to redeem a GOG (gift) key.
That appears when you click on the GOG ball on the top left and choose "redeem key".

that sentence reminded me of a certain paperclip...

Is the Code box four character long? Do you get it when trying to log in?
If yes, it is probably the two-step authentication, and you would want to check your mail inbox.

EDIT: It is Probably the redeem page.

Yep, it's the gift code redeem page. What you have on the screen is a game client / launcher, GOG Galaxy. You appear to have Gwent already installed (or at least have an entry for it).

To launch a game that's present in the left-hand panel, click on its title there to open the game's catalog page then click a prominent button (Play or perhaps Install) in the main window. I don't have Gwent installed so I'm using Spacechem as an example (see screenshot).
Alternatively, right-click the title in the left-hand panel and choose "Play now".

To launch a game that's not present in the left-hand panel (one you bought, or a free game), click Library and find it there.

Guessing everyone here is not South Park fans. Miserable gits.

Viva la resistance.. death to Galaxy!