My son didn't show much interest in chess at an early age, and I didn't push it on him. But there's a chess club at his elementary school, and he joined it in October, as a first-grader. The club only meets once a week for an hour, with instruction for the first half hour and then actual game play for the rest of the hour. I've tried to attend and help out as much as I can.
He's become more interested to the point where we play a bit almost every day, and he's shown marked improvement. At first, he was just learning the rules, so I wasn't sure what handicap I should give him. My goal was to never have to let him win, so I had to come up with a handicap that made it somewhat even -- where we'd each win between 25% and 75% of the time. If he lost too much, my feeling was he'd get discouraged, and if he won too much, well, he needs to learn to handle losing, as well. It also helps teach him to learn from his mistakes.
So, my first attempt at handicapping was to simply use a chess clock and give myself a very short amount of time and him a very long amount of time. That didn't work at all. Beside the fact that the time handicap wasn't enough, he didn't even use the handicap. With young children, if you move fast, they'll move fast. It's like an impulse. Even if they have an hour on their clock, if you start making blitz moves, so will they. It's frustrating to watch. Most of the games in his chess club are G/20, but the beginning players especially have their hands cocked over the chessboard, waiting to make the first move that comes to mind, as soon as their opponent finishes their move (often before they even hit their clock). We have to constantly tell players to slow down.
So, seeing that the time handicap wasn't working, I cleared out all my pieces, leaving myself with 8 pawns, a king, and 2 bishops. I still won - easily.
So, for a while, I started teaching him endgames. KQ vs. K, KRR vs. K, KR vs. K, and KP vs. K. As you may have noticed, this prompted me to make videos for those situations. Hopefully they'll be useful to other kids (or adults) learning chess. I'll probably make more.
I think teaching endgames to kids is important. There's a sense of accomplishment that comes with checkmating your opponent. Also, there's the practical advantage that it provides over your opponent -- most kids who haven't otherwise been taught have no idea how to mate with KQ vs. K, let alone KR vs. K. In his chess club, I see lots of moving back and forth, aimless continuous checking, and oh, so many stalemates. My son had problems with stalemates, as well -- not so much when he had KQ vs. K (although he did at the very beginning), but when there were lots of other pieces on the board. Kids love to get multiple queens on the board, but the inevitable result is a greater chance of stalemate.
Eventually, I got back to playing games against my son. I decided to pare back even more, limiting myself to KNB and 3 pawns, all on the kingside. I don't think that lasted long before I moved up to KNR, and then KNBR and 4 pawns, all on the kingside. Basically I was playing with half my pieces. It took him a long while before he could beat me with this setup, but eventually he got it. Very quickly he realized that his favorite thing to do was to bring out his king pawn, then check me with his bishop so that I couldn't castle. I was helpless to stop it because of my lack of any queenside.
Now, we've moved up to the point where he can usually beat me if I have everything except my queen and queenside rook and the two pawns in front of them. I've also started giving him a time handicap, where he gets 20 minutes and I get 7 minutes. And he's learned to use his time more wisely.
I worry sometimes what the handicapping is teaching him, since he will happily trade his rook for one of my knights or bishops and still have no problem beating me. But I figure it's only temporary until he plays well enough that I can get away with just the time handicap, perhaps combined with a small material advantage. I look forward to the day when I can play against him with a queen and both my rooks -- do you know how frustrating that is for me, wondering how in the world I'm going to win without them?
So, what about openings? I haven't taught him any. My theory is that he can learn the rest of the game first, and if he gets serious, then he can study openings later. To me, that's more fun, anyway, and the way I learned. I never studied the openings until I got better, counting on my superior endgame to help me win more games. I do, of course, teach him basic opening principles. I'm constantly asking him where he plans to post his knights and bishops, and how his pawns can end up blocking his own pieces in the opening.
When we don't have time for a full game, I'll play little exercises with him. For example, K and 6 pawns vs. K and 4 pawns. Or KQ vs. K and 5 pawns. I think it's important to give him a variety of situations, and things where I can give him instruction as we play.
I hope this have given you some ideas on how to teach kids chess. Feel free to let me know if you have other ideas. And I'll write more and post more videos in the future, probably some tactics videos in the near future.