Basic knowledge:
Tetris
It's videogame 101. A simple puzzle game about falling shapes and clearing lines. Tetris is timeless because its formula does not need to be reinvented in order to be fun. There is no end, so the main objective is also the only one. It's simple, it's intuitive, and it's remarkable. It's hard to find a videogame more videogame than Tetris.
Super Mario Bros
It's videogame 102. You control a small, agile avatar that runs and jumps. Anything you do in the game is based on running and jumping (and throwing fireballs). It teaches how movement is important and the mere fact you can control an avatar character and do different stuff with it can be incredibly fun. It shows less is more and there is no need for a ton of things if you're making an accessible game.
Doom
It's videogame 103. Shows how speed, positioning and quick reflexes can make a game incredibly engrossing in a 3D space. Shows how enemies should behave and how frantic action paired with secrets and knowing your audience can lead to great success. Doom's worldwide success was no accident. Knee-Deep in the Dead is still a masterclass on how to make a fun first person shooter.
EarthBound
Every single person in this world should play EarthBound. It shows how a game can be
kind, and how humanity, instead of escapism, can make the soul of an adventure. It shows how dialogue, charisma and events can have real effects on one's perpection. It's a game that made me a better person. I can't say that for many other games. Similar recommendations, but for different reasons: Dragon Quest V (teaches the importance of narrative through a personal journey), Chrono Trigger (shows the importance of masterful pacing and uplifting tone), Skies of Arcadia (same as the previous one, minus the perfect pacing), Panzer Dragoon Saga (all of the previous ones; plus, it's unique) and Pokémon (acessibility, incentive for playing, making a game for all ages, interaction with other players). Also, the obvious ones: Mother 3 (EB with more narrative and fleshed-out characters) and Undertale (influenced by both games, shows how kindness, good writing and subversion of expectatives can make an experience remarkable).
Advanced courses:
Super Mario 64 and Super Mario 3D World
Same as the original SMB, but in a 3D space. Super Mario 64 shows how a game should be fun simply by moving the character around. It also teaches basics about 3D cameras and how they sucked. Super Mario 3D World, ironically, is actually this formula
simplified so as to be an actual 3D conversion of the game style of the original, 2D Super Mario Bros games. It also teaches how 3D space can be manuevered with a fixed camera
and multiplayer and still work despite all that. Both are incredible genius games, which focus on fun first and foremost.
Resident Evil 4
It is the quintessential third-person action game. Masterful pacing, encounters, player feedback, tension, and even tone. Almost every single thing you do in this game is enjoyable, down to the most silliest and most irrelevant ones. No encounter plays the same. Even
reloading is fun. There are few games that come as near perfection as this one.
Runner up: Resident Evil 2, for a near-same example on a simpler hardware.
Osu! Tatakae! Ouendan and Elite Beat Agents
I sincerely consider the Ouendan/EBA series to be quite possibly the best IP to come out on the previous decade. While Guitar Hero had more mainstream appeal, and the iNiS games didn't really set the world on fire, I think it's remarkable how the games' unusual control scheme is so simple yet so acessible to the point where I've never, ever seen a player think "Ok, I've had enough, I don't feel like trying the harder difficulties". Ouendan and EBA have a steep learning curve that makes even people who don't give a shit about videogames go insane on making high scores, perfect runs and keep improving themselves. Seriously - throw this game at anyone who never played a videogame before, give them a while to understand how it works, and see it they'll want to stop. Stupid fucking spinner aside, these games are absolutely genius.
Rez
It's art. Teaches how videogames can be sinestesic experiences. Also recommended: Space Channel 5 Part 2 and Rhythm Tengoku (games with incredibly simple control schemes that are immensely rewarding by the way they're presented).
Really fucking advanced courses:
Metroid Prime
Teaches the importance of masterful design, exploration, sense of progression and atmosphere. The more I understand videogames, the more I like this one.
Dark Souls
Shows the importance of risk vs. reward. Shows how games can be engrossing if the players feel there's something on the line. Shows how good world-building and a sense of tension create an engrossing experience. Demon's Souls, is predecessor, is also excellent, but I like how this one makes it with an interconnected world.
Bayonetta
It teaches what an action game is.
Vagrant Story
Teaches the importance of a great script, art direction and cutscene direction. While Metal Gear Solid is the obvious answer to this, I'd go with the underdog, as it also shows how a steep learning curve can make the players more enamored with the work if they go deep enough to understand it. I hold VS in a high regard that almost no other games match, save for other Matsuno works, such as Tactics Ogre.
Personal recommendation:
Wonder Boy in Monster World
It's the simplest, most colorful, more fun introduction to a Metroidvania one could ever play. It encourages exploration, trial and error, and lateral thinking (considering what was standard at the time, in the system). If I had to recommend someone to play a game to learn what games are, this would probably be my pick.
Ok, I guess that's enough.