If it begins with GC2- and that's the only serial that came with it, you should be able to trade it in at https://www.galciv2.com/paradox. If that doesn't work let me know.

Yep, that should be it. Grab stardock central from the downloads page, install it and log in with your forum account--that exchange page should have already registered it for you if you, in which case GC2 will appaer under 'Purchase Products' in the games section.

Be sure you uncheck the 'show pre-release versions' option at the bottom first, as otherwise you'll end up with the beta patches which are best avoided by new players.

If it doesn't show up there, you can register the game by going to tools menu->register product, pick games on the left then find GC2 in the drop down list and put in your email and serial.