How do I download a game I just bought?

Go up to account -> library, or just click library if downloading by Galaxy.

Explained here:

https://www.gog.com/forum/general/how_to_install_worms_wmd/post3

you forgot to include the third option for the gog downloader
it might not be maintained anymore but its still working

Or gogrepo.py, or lgogdownloader, or Down-It-All, or even downloading the installers through the Galaxy, or or or...

Yes there are various ways to download the offline installers, but for simplicity's sake I'd rather present it like this:

1. You can use the Galaxy client to download/install your GOG games (a bit similarly how Steam or EA Origin clients work).

2. You can download offline installers for your GOG game and install the game with it.

For this latter option, there certainly are several ways you can download those offline installers (including your web browser, the Galaxy client itself (using its "Backup" option or whatever it is nowadays), the old unsupported GOG Downloader client, gogrepo.py, lgogdownloader, the third similar mass-downloading tool whose name I don't recall etc.).

It doesn't really matter in which way you download the installer files in this second option, as long as you do.

Furthermore, my personal reasons for not advertising the old GOG Downloader client are:

- It is unsupported and superseded by the Galaxy client, which you can also use to download the offline installers.

- I personally felt the GOG Downloader was a buggy mess, I had failed downloads with it quite often which I couldn't even resume.

- Considering the OP doesn't know how to download GOG games with his web browser, telling him to use GOG Downloader will only add to that confusion as the GOG Downloader is not that simple to use because you have to log in both to it AND your web browser, and you are supposed to click on the Downloader links on your web browser so that they are caught by the GOG Downloader. If he had problems understanding how to download games with a web browser, it is even harder with the GOG Downloader (for a first-timer).

^ then its high time the OP learns