ZollOrr: Thanks, Ancient-Red-Dragon, for letting me know about the "offline installer" and that I don't need the internet to play the games (possible exceptions noted). I have a few desktops and laptops of various ages which might give me an edge as far as getting a game to work.
As others have already noted, some games may still have issues on newer hardware and/or Windows 10 (or sometimes even Windows 8 or 7, but in my experience 7 has much better track record with the problematic GOG games).
Old games that run either on DOSBox or ScummVM don't normally have any real issues, so e.g. MS-DOS era games should be all pretty much ok. It is mainly the early, Windows 9x era games, that have the most compatibility issues (unless they also run in ScummVM; like Myst 2: Riven started using ScummVM at some point and that allowed Windows 10 users to play the GOG version as well).
So if you are thinking of buying some old game, first check its gamecard. If Windows 10 is listed as one of the supported operating systems, that is a good indication the game (more) probably works on your modern systems, and what is even more important, if you fail to get it to run on Windows 10, you are eligible to get support from GOG to get it running there, or get a refund. (Then again nowadays they might give you a refund anyway, even if you were trying to run a non-supported game on Windows 10, not sure...).
Another thing is to visit the game's GOG subforum. If there are serious issues to get the game running on modern PCs, there are normally many threads about it, also with tips how to get it to run.
For instance, I've been lately testing to get the GOG version of
Dark Reign to run on various PCs of mine. Windows 10 is not supported, and many people seem to have lots of issues to get it to run there, possibly due to the inert (not removed) copy protection that the game still seems to have, and which may act up sometimes.
The best success so far to get it to run for me has been:
1. A genuine Windows XP machine. There it seems to generally work ok, but with one caveat: you may not have any windowses open while you run the game. If you do, at least I will get that "no CD found" copy protection error in the game and it refuses to run, even on Windows XP.
However, if I make sure I don't have any windowses open (not even File Explorer) and start the game straight from the Start menu, I still get that "no CD found" error but the game still starts and runs fine. Even the in-game music works.
2. Linux (Mint) with WINE: The GOG version seems to install and run fine there, I don't even get that "no CD found" error as I do in Windows (XP). However, the ingame music doesn't play at all in Linux, but otherwise the game seems to work fine, with sound effects, the video parts work too etc. (The missing music has been reported also by many Windows users, but it works for me in Windows XP specifically.)