So, I got a new computer with with the Windows 7 OS and one of the first things I've done is install the Fallout Trilogy disc on it, but there's weird wonky colours. What can I do to fix this?

IIRC, the best way to fix that issue is to run the games on Windows XP compatibility mode. Right click the game --> properties --> compatibility. If that doesn't work, also try launching the games in administrator mode. It has been a while since I fixed that same problem on my computer, so I can't remember if there is any other fix out there.

What I usually do is change the theme to Windows Classic and then leave the "Adjust Screen Resolution" tab open which you get to by going to the control panel.

I haven't the faintest clue why this works (leave the screen resolution screen open when you start the game), but it does.

Install the high resolution mods for both games (http://www.mediafire.com/download/ua4bj85mf12xj3g/Fallout1_High_Resolution_Patch_4.1.8.zip, http://www.mediafire.com/download/7uaa8r5600r9n92/Fallout2_High_Resolution_Patch_4.1.8.zip), and in the new in-game option menu "screen settings" change the color depth to 32 bit.

I can confirm that the above suggestion worked for me.