When I try to start this game, the screen goes black for a few seconds, and then I get an error message. (See Attachment)
I have the following system:
Wndows XP Pro Service Pack 3
AMD Athlon 3800+ X2
3 GB Ram
ATI HD4670 512 MB Graphics Card
Onboard Audio
I tried disabling sound acceleration and Directdraw in my Quicktime settings as suggested in other threads, but it did not work.

You could try turning hardware sound acceleration off from directx as well. Go to start menu then run and type dxdiag. Once that is open go to the sound tab and just move the slider all the way to the left to turn it off. Remember to put it back when you play more modern games.

I tried doing that. It did not work.

Could try updating quicktime to the latest version then and hope it works. If that doesn't work then maybe uninstalling quicktime and installing a version of it that is closer to the time septerra core came out might sort it? Assuming its possible to find a version of quick time years old now.
It seems the gog version needs quicktime though so they should of packaged that in the installer. So you could try uninstalling the game, uninstalling quicktime completely and then reinstall the game and hopefully quicktime will also install or ask to be instaled with the game. If not you could try finding an old version on google.

I tried doing those things, and it still doesn't work.

I had the same problem and what I had to do was go into the Septerra Core directory and there should be a file called Quicktime.qts and I think its outdated. So what I did was go into my recent quicktime install folder, and find the updated quicktime.qts, copy and paste it over to the septerra core directory and then the game worked!.

I was told to do that at another forum. It worked, but now the game skips all the movies. :(