Purchased Stardew Valley for the first time, using GOG (which I've never used before). I was able to click install and waited about 30-40 minutes until a pop-up saying: 580 MB download etc, along with T&Cs, where it's being installed to. I then clicked the second install on the pop-up and it closed the pop-up. I waited for a few minutes, but nothing was happening. So, then I clicked on Stardew Valley again and saw that the install button at the top was greyed out and I could not click it.
I've looked through their help page on SV, checking advanced program files, running it as administrator, and adding it to my 'firewall' service in games. But it's still greyed out and unclickable.
Any suggestions?
Thank you!

Are you trying standalone installer, Game+Galaxy installer, or did you install Galaxy and are trying to install from that? Each is a bit different, though you shouldn't have issues with any of them.

Restart your system [which is different than "shut down and then turn back on"; restart in modern OSes does more, believe it or not], disable your antivirus (GOG's a trustworthy source), and try again. Re-enable your antivirus.

Welcome. I guess you fixed it (see attachment). For future reference, clicking Install should immediately queue up the game in the Downloads section. If you get a spinning circle that doesn't seem to go away, disable IPv6 and restart your system.

Edit: Also, you can use offline installers instead. These can be downloaded using a web browser, from your library. Some games don't have any online bells and whistles like achievements and cloud saves. In those cases you're only missing out on playtime tracking by not using GOG Galaxy.