I'm in season 3 of teaching my niece (10) and nephew (7) to ski. Some things I've learned.
Start with group lessons. Understand that for a kid, beginning lessons are as much, if not more, about the other kids as it is about the instruction. Kids will help other kids get over mental hurdles and model success in a way you simply cannot.
I'm a firm believer that kids can do a lot more for themselves than many parents I know give them the chance to do. It's all about motivation and expectations. I know they can untangle themselves, get their own boots on, put their own skis on when at an angle, walk uphill in skis, etc. I don't do any of that for them and I constantly have to beat off other parents who try to help them when they don't need it. Don't get me wrong, I appreciate the sentiment and helping spirit, but I want the kids to get the practice doing something I know they are perfectly capable of doing themselves. It also saves me a lot of stress not having to do all the little things for them all the time, and that pays off in buckets when I am working with them on the hill and stress free. Just as important, you have to know their limitations and know when it's time to step in and help. Keep them from losing spirits when failure is nearly guaranteed.
Teach them to fall! I remembered this from when I was little and learning to ski. Once I learned how to fall properly, when I knew I was getting out of control, was when I really started to learn how to ski. I could take a chance, and if it didn't work out I could fall before things got out of control. To a kid that can be a powerful tool. It made a huge difference for the two I take.
In the same vein as my last two, I never really used any tools like harnesses or poles to assist them. The ski school may have, I am not sure. The exception to that was the first season with the youngest (4 at the time) who used tethered tips. He simply lacked the leg muscles to hold a wedge without them. Beyond that I taught them to fall and since then they have always been the ones in control. The closest I get to taking away their control is to steer them by cutting off their fall line when I feel it's necessary (I only recommended this if you're good enough on skis to handle the unexpected when skiing close to others). But now that I have mentioned it, if you are able to do it safely, I actually found that to be a very effective way to keep kids skiing in your comfort level. It can even be a great game, dodge the uncle!
It's all about the fun. If having fun, kids will learn and improve on their own. If not having fun, no amount of effort on your part is going to make them learn. Progression picked up for both of them when I spent more time focused on making sure they were having fun than instructing them on how to improve. A lot of this ends up being making sure they are warm and dry and keeping them loaded up on high protein snacks. I still instruct, but I try my best to make sure it doesn't detract from the fun, which leads me to...
Don't be afraid to teach them the theory. Each time we step out on skis I go over theory with them. We look at how they curve on edge, how they load and unload, edge control, etc. I show them on my skis so it's nice and visual. Kids have no problem understanding this stuff. Just keep it short. Then I remind them regularly while skiing. Pick a key word or phrase and always use it when discussing the topic. Then it's as easy as repeating the word/phrase when it's time to remember the concept on the hill. I've had a lot of success with this. It doesn't interrupt the flow of the fun while they are skiing, and it becomes effective rather quickly.
Be careful not to get them into something out of their skill level. I made the mistake this year of taking the girl up early season. I knew she could ski the angle, she skied similar runs the year before, but it was firm early season snow with few runs open and swarms of people. Taking her on the lift was the bad decision, but once made, you live with it. 1/4 way down the run (and almost 2 hours later) she is in tears, I'm a little freaked out at the idea of carrying her with the crowds that dense. No one, including ski patrol, really want to take her down in a sled. Eventually she calmed down, and in little bursts between the masses coming down the hill, she skied out just fine. Not the worst experience I've seen parents have, however, that one experience set her back a full year in skiing and there's no question in my mind it's mental. That experience scared her. She no longer is willing to fall, which is not helping her learn to ski at all. Live and learn.
Last, I take the right skis. I take park skis that I can manipulate without a thought. I want to be nimble like a fox, not burly like a bear!
I am no expert, by any means, this is all new to me, and has been a bit of trial-and-error. However, I have been happy with their progress, and have certainly fostered a love for the sport in both of them. It's been as much a learning experience for me as I'm sure it has been for them, and I doubt that's going to stop anytime soon. I look forward to hearing other peoples experiences.