AB2012: As others have said, offline DRM-Free installers is pretty much GOG's "killer feature" over Steam (certainly for single-player games).
Hover over your username at the top -> Games -> Click on desired game -> Underneath "Download & Install Now" (Galaxy), there's another option called "Download Offline backup Game Installers". Depending on how large / small the game is, you could have one or multiple files to download, but click on them to download them to wherever your browser's Download folder is, and once there double-click to install like any other program download via a web browser. They'll then wok without Galaxy and all the Cloud save, etc, stuff.
As for where the local saves will be stored, it varies from one game to another, but
PCGamingWiki is a good resource that details save / config locations.
I know people mean well when the suggest offline installers, but they aren't always the best solution in every case. OP never indicated they wanted to have a backup of their game.
Simply playing offline is as simple as not using Galaxy, instead launching from the games exe file. Getting rid of Galaxy would simply be uninstalling it, your games would still work. And getting rid of cloud saves is simply going into Galaxy's options and turning it off.
So by suggesting offline installers you could be wasting somebodies time and bandwidth to get the same result they could get without offline installers without making that clear to them. If OP kept Galaxy installed then followed this to get offline installers, then installed them, the end result would be the same. Galaxy would detect the files, so it would not really help them at all.