Hello, I bought the game on steam, as well as Gog because I wanted to support it. Now I was wondering how exactly would the steam version and the gog version of the game play with each other. I am quite confused on this matter. If they really can't play with each other I wonder how to transfer my save to the Gog version. Please help.

Just open the game's main folder and you find a "Saves" directory. I guess that is everything you have to copy over.

Anyway the Galaxy and the Steam lobby is incompatible, even the store page says it. However if you run the game without a client (starting directly from the folder) it enables you to specify a server by entering its IP address. While I'm not sure if the GOG and Steam version are identical but you can try hosting a LAN server like this then using the IP to connect with the another. Of course without any of the clients running.