How do I fix this?

A screenshot might help people understand your problem

That looks bad. It's as if your graphics card is trying to use a 16 color mode

Install Killap's Unofficial Patch (or Restoration Project):
http://www.killap.net/
...and change graphic settings for D3D:
https://www.gog.com/forum/fallout_series/fallout_2_screen_is_black_on_launch_and_the_resolution_is_always_1024x768/post3

https://www.gog.com/forum/fallout_series/fallout_2_terrible_slowdowns_on_windows_10/post2

Works?

The patch fixed the colors but now the game area's half covered with black and looks something like this
Screencapping doesn't work now so I had to edit a picture to make it look like how the game looks

This is a known problem when trying to run FO2 under WinXP. Using the Win98 compatibility mode (plus installing the 1.02 patch when necessary) solved it .
But I have no idea if this works for newer Windows versions or the GOG version of FO2.

The rainbow colors?
I've never seen that in any game in XP before, it's only been a problem with windows 7 =s
Unfortunately it's as if every game I want to play most has this bug and aren't in dosbox so I can't change it to 3D3

I can't seem to reply directly to posts either, the post box just keeps loading forever

I got my backup version from 2009 and it's still doing the color thing, it never used to do this before =s

All of those initial images have less than 20 colors, as if the full palette wasn't realized and it was using only the 20 "reserved" system palette entries.

Try it with DDrawCompat. It does palette conversion in software via GDI, may have better results...

Edit: Ah, I just noticed you may be using Windows XP (are you?) in which case this probably won't help. (Newest version definitely wouldn't work as it links to dwmapi.dll, older versions are simply unlikely to work as I never bothered to maintain support for such a legacy OS).