Hopefully things will go smoothly for you to get the game. But if not, here's a guide to help you..
When you purchase the game you will receive a link to Stardock Central. Download that.
Make sure you are running 2.29 or later (if you've been running Stardock Central for awhile, double check in the about box that you're not using an old version like 2.24).
If you're downloading the game and it's slow, you can try a different server. From Stardock Central go to tools->Settings and the server is in the "Internet" area. We have several different servers.
If you used a different email address to purchase the game direct than what you used to create an account here, Stardock Central may get confused. If that happens, email firstname.lastname@example.org with the subject "GalCiv Account Issue" and they'll take care of you. If you receive no response within 1 hour, post here and someone from Stardock will likely see it and take care of you here.
The core game download speed will be slower than the multimedia/movies/tutorials because those are mirrored. The core game is regularly updated so we aren't able to mirror it as much. You should still get pretty decent speeds (and a reminder: try a different server if you find downloading too slow or if you get a time out -- bear in mind, there's a LOT of people downloading the game).
If you have problems related to actually obtaining/installing the game, post them HERE in this thread.
NOTE: Some anti-virus software will cause the installer not to work (i.e. it'll block Stardock Central from extracting/installing the game). If you run into an installation problem, try disabling some of your anti-virus setup temporarily to see if that resolves it.