MightyPirateScipio: As a side note, my original idea was to go into the GoG store and just buy the best-seller or the highest-rated game from each section, but I noticed that the rating system in particular might not be a good way to go about it. Generally, I just thought asking the community of long-time gamers might be a better source.
Like most other things genres are somewhat blurry. In terms of the biggest classics that have aged well from each of gog's genre bestseller pages I'd say:
Action - Beyond Good and Evil, Thief/Thief 2 and Chronicles of Riddick (although most would call the latter a shooter).
Adventure - Monkey Island, Longest Journey, Broken Sword. Maybe Gabriel Knight and Grim Fandango but they're still on my to do list. Tex Murphy is enjoyable but you have to persevere with the controls for the early ones.
Indie - Not a genre.
Sports - GoG doesn't have any of the big games.
RPG - Baldurs Gate II, Deus Ex, System Shock 2 maybe the original Witcher or Planescape Torment.
Shooter - S.T.A.L.K.E.R - Shadow of Chernobyl. Most big shooters are too new to be on gog or too multiplayer focused to be something you can come to later.
Simulation - Sim City 3000/4, Rollercoaster Tycoon.
Strategy - Gog is missing the big sellers like C&C, Starcraft, Age of Empires, Warcraft etc. but there are a lot of very good lesser selling options like Galactic Civilisations, Startopia, Total Annihilation, Men of War & Space Rangers. I'm also a fan of Europa Universalis III/Crusader Kings but they are less mainstream.