I don't know how to describe this. I have almost no problems with most games in GOG except for somee versions running better than others.

Anyway, this is the problem: The start-up cut-scenes run normal, with no color problems. But once I reach the main menu, it "freezes". It doesn't show up the menu or my mouse and it keeps running and I can get out from the game normally. However, it maintains the last "image" it took from the last video, even when I try to skip it all quickly.

Moving my mouse doesn't reveal anything, in fact I can't do anything. I've been trying several "rainbow colors" solutions but so far nothing.

I have to admit, this computer is rather old intentionally so I don't have these sort of compatibility issues with old games.

I need help. Maybe the answer is out there and I haven't searched well enough.

Which fallout are you having this problem with?
If it is Fallout 2, try opening the game by just going to the exe in the file it is in, not using the desktop icon. Fallout 2 seems to be a little buggy for more modern OS's un-modded. Otherwise, I've seen Killap's Unofficial Fallout 2 patch and Restoration Project are highly regarded, and address a multitude of issues and restores/adds content that was cut from the game.

For Fallout 2, I'd recommend Killap's Unofficial Fallout 2 patch or Fallout 2 Restoration Project as well, but install only one them, as the Restoration Project is basically the Unofficial Patch plus extra content.

Download link: http://www.killap.net/fallout2/web/Downloads.html

For Fallout 1 try Fallout FIXT:
http://www.nma-fallout.com/showthread.php?194562-Fallout-Fixt-alpha-6-7-3-June-2nd-2014
It comes in different versions, fixes only or fixes plus additional content - whatever you prefer.