How do I play online? I got the serial key and pasted it in the authentication thing, but it just says "Authentication status: Checking" and then "Authentication status: offline"

You've probably either refunded or found the fix by now, but I thought I'd answer anyway. In the install folder, there are two executables. defcon and defcon_offline. By default, GOG uses the offline one, but there is a button in GOG Galaxy to use the multiplayer version, not quite sure why it's not default though