Like, I can change the color by doing the [COLOR=Black] thing in my signature, but I can't just click on the "A" with the line on it. It doesn't show up with the color options. It just shows a small white bar when I click it, and that's it. No matter what I do on my computer, doesn't matter what browser, it just won't work.

Yeah that happens with everyone I guess. Just type another color in the part which says "black" now
PRetty sure the color list's color names work too http://forum.toribash.com/tori_colors.php

i think you have to select 'go advanced' which is next to post quick reply, that may be the problem.

Yes, it's a bug with the forum. The dropdown list doesn't work in advanced options/posting mode. As a workaround you can go to some thread and use quick reply window to write what you want in the signature (color dropdown list works in that window) then paste it into your sig.