I didn't buy SV on steam, instead I got it from GOG. But it seems there's updates, how would I go about getting these?
No, whenever GoG Galaxy pushes an update, it forces you to redownload the game files - The way I see it is the game and patches are so small it's not worth making an entire standalone patcher for it. So you will just download the game again and the files will go where they need to go. Your save is fine, patches don't touch the '\AppData\Roaming\StardewValley\Saves' directory.