How I teach people chess
Almost every time I visit a chess club, somebody is learning chess. Maybe some temporarily embarrassed GM has brought their kid along and they're curious; maybe it's the other way around and some poor six-year-old is trying to impart what they know to their curious mom. No matter what, it's almost always a barrage of information. First they go over how the pieces move, then the special moves castling and en passant, then the dynamics of checkmate and stalemate, then a few principles like center control and maybe some forks and pins and skewers, and then they'll get into sorry are you still reading this sentence my eyes glazed over halfway through. If they're merciful they'll skip "J'adoube" and touch-move before getting into the first game, where they absolutely cream the newcomer, who may or may not still be curious about the world of chess.
Some games can get away with being taught like this. If some Commander enjoyer infodumps about the rules of Magic to somebody and then creams them with their totally-a-7 Scion of the Ur-Dragon build, the noob will probably try again because of the pretty cards and the trust generated by your friend saying they spent hundreds of dollars on the game they talk about constantly. If you learn go and then get annihilated, it's fine because there really are like two rules in that game, so you probably discovered a tactic and want to know more. Chess is a great game with a pretty unique appeal in the market (a multi-millenium history, the satisfaction of perfect information, a global competitive scene) but it doesn't have the same viral quality as Commander. If somebody learns literally every rule of chess and then has a bad time in their first game, they will probably learn to be content merely watching their eldest son play a game that, unbeknownst to them, is pretty good when you get the hang of how the horsey moves.
That, dear reader, is why I have developed a theory of how to teach people chess. I've put it into practice a couple times, including with my own partner. I haven't rigorously compared it to other ways or anything, but I feel confident that I have a good way to keep somebody entertained while they learn the rules of chess in a 1-on-1 setting.
How the horsey moves
First things first: high level overview. You take turns moving pieces, threatening and capturing, and you win by trying to capture the king. A lot of this stuff is familiar to them through osmosis, but you should still say it, or at least ask a question like "So what do you know about chess?". Depending on my newbie, maybe I'll say something about the joy of perfect-information 2-player games in the age of Magic and Farmville.
The brief summary segues oh-so nicely into talking about piece movements. If you talk about the king first, you can set him up on the edge and show an example mate. I usually show a two-rooks mate and a king-and-rook mate, because it's easy enough to mention in passing how the rooks move and easy enough to see that the king can't move away. With the kings and the rooks, you can now define check and checkmate. Ideally you'll start a ladder mate, ask them to finish it, and boom-- baby's first checkmate. I will absolutely not talk about stalemate yet if I'm not prompted, because I have priorities. Also, even though I'm talking about the king and rook, they don't need to know about castling quite yet. We structure information in a way where they'll get there eventually.
Bishop, queen, and knight come next. I try to be hands-on here, especially with the the knight: make them place a pawn everywhere the knight can move, then make the horsey do a fork or journey from a1 to h8 without being capturable by a queen on h8. Then I ask them to move the bishop from a1 to h7, teehee.
The pawns come last. I talk about the one-square move first, then I talk about how centuries ago people were bored and decided they should be able to move two on their first move. This is an oddly specific way to present it, but it's important setup for making them remember en passant later. Then I talk about how pawns capture weird, briefly mention promotion, and that's it. They know how all the pieces move, they know what checkmate is. They don't know all the rules, and they don't know any heuristics like center control or rapid development or pawn chains. If you put them in front of the white pieces and asked them to follow their heart, they could very well open a4 into Ra3. It's time for their first game.
The Lone Warrior
I set up the board. They get a full board as white; I get a king and queen as black. I tell them this is very scary, because I'm going no mercy and my queen will start attacking them from turn 1, but I also let them in on a secret. "If you make sure all of your pieces are always protected," I say, "my queen can never take anything without getting taken back. And I only have one piece, so if that happens, I can't win anymore, because my king can't checkmate you on its own." And just like that, the game starts.
I can't express how fun it is to terrorize a chess noob in a completely safe, fun way for both of you. It's like the opposite of losing a winning endgame against Stockfish. As long as you don't smothered mate them, you can do whatever you want. Don't get me wrong, at first you should tell them "Raawr, I'm attacking your f-pawn" or whatever, and throughout the game you should gently coax them into making random discovered checks. But you can also absolutely just gobble their rook when they hit you with that g4. You did, after all, say there would be no mercy. It will be that much more satisfying for them when they eventually coordinate their pieces and stumble into a crushing tactic, which you will point out and name for them (just as you have been saying "You see this? This is called a fork" before gobbling a minor). You will probably lose your queen by force, if not by losing track of tactics, and your ward will be left with a hilariously stacked endgame.
This is secretly the point. Now the newbie has the perfect playground to learn about some super-basic endgame technique. This is the moment where you explain that, congratulations, you can't lose anymore, but there is this thing called stalemate you have to make sure doesn't happen. Now is also the time to explain pawn promotion more fully. Explain out loud when your king gets trapped in invisible rook boxes, gently advise them to get pieces in on the attack, explain how you're trying to escape to the center but their queen is fencing you in. Don't let them stalemate. Ideally they win by just continually throwing new pieces at the problem and mating by accident, which looks spectacular and clever and coordinated. Do a drawn-out "Hmm... I can't move here, I can't move here, I can't move here, I can't capture this or block this...", and congratulate them.
There are a few reasons to play Lone Warrior for your first game, the most obvious of which is that I'm smiling just thinking about it. It feels good for both parties-- they get to face off against a frankly terrifying threat and inevitably stop it, and you get to eat hanging rooks, all in a safe fun way. The second is that they get some good practice moving pieces in real situations before you talk about the Weird Rules and basic strategy. They also discover tactics in real life-- most of the time, you'll at least threaten a fork, and they'll pin your queen (and discover that it's good to trade bishop for queen, a surprise tool that will help them later). Perhaps most subtly, though, they went through an endgame. A lot of noob-v-noob matches enter a point of confusion when it's time for one player to not accidentally stalemate-- a little emphasis on how your opponent's rook is boxing in your king goes a long way.
Aftercare
There are only a few more things to do before letting your trainee loose ("one more really important thing, and one more silly rule, and that's it-- you know everything"). The only ones that are strictly necessary are the Weird Rules: castling and en passant (and I guess promotion if you didn't see it in the game). It sucks that you have to teach en passant, but the ability to say "That's it, you know all the rules, everything else is you journeying" is a real privilege. It's made easier when you frame it as a hitbox glitch introduced by old-timers letting the pawns move fast on the first move (remember when you said that). Castling is actually important, spend some time on that one, explain how nice it is to have your rook in the middle and your king sequestered by some pawns you don't care about.
But why would you want to castle, you might ask? Well, that's why you segue into heuristics. The real basic ones are nice to know. Castling is OP, move your pieces forward, control the center, know the relative values of pieces. They probably realized in your game that center control is pretty good, use that. They also probably discovered a tactic, so you might use it as a hook to help them fill out their Pokedex with forks, pins, skewers, and discoveries. Double checks, maybe, if you're feeling kinky?
Endgame
Chess is a pretty simple and noob-friendly game. It's not perfect in that regard-- no noob in their right mind would independently formulate the piece values. But you can explain all the rules in one sitting, and you can make them all stick if you put a minigame in there. A beginner can't even choke by walking into checkmate, since it's illegal rather than instantly losing (and if you don't think that's special, ask a Magic judge what they think about about declaring Vehicles as attackers in tournament REL). Your noob will still be confused and scrambling, but they can certainly go off into the wild and meet new friends over the board. They probably stand a fighting chance against intermediate beginners if they're lucky, since they've had experience formulating a plan and protecting their pieces.
The most important thing is this, though: Even if they get creamed their first game, they probably won't quit chess forever, because they got a taste of what it's like to learn the game. That's your goal.