HereForTheBeer: So you have three options, all mentioned earlier: Galaxy, the official (no longer supported but it has worked fine for everything I've thrown at it) Downloader, and plain old browser download.
For simplicity, I'd rather say there are two ways:
1. Tell Galaxy client to download and install the game (and any patches) for you. This is the "Steam-like experience", but apparently is failing for the original poster for some reason.
2. Download the offline installers (and DLC installers and patches, if applicable), and install the game yourself with them.
There are various ways to download those offline installers:
- Galaxy (under "More", or was it "Extras"?), you can also download those aforementioned offline installer files, and then install them manually, without using the Galaxy client. You can later import the installed game to Galaxy, though.
- With your web browser (you can use any web browser downloader plug-ins too, if you wish)
- The legacy GOG Downloader client (not supported anymore, and frankly I don't suggest that to beginners, it is even more confusing to use than Galaxy... I mean, you need to log in BOTH through the client and your web browser, and then initiate the downloads with your web browser... plus I quite often ended up with failed downloads with it, even more so than with a mere web browser).
- A number of user-made tools, like gogrepo.py, lgogdownloader, the third script someone made whose name I don't remember... Quite often these are meant for specific tasks like downloading ALL your GOG game offline installers, not just for one game, but for all. For mass downloads. Maybe not the best solution just to download and install e.g. Pillars of Eternity.