This is an area of active thought and and evolution for me as I figure out better ways to do things. Ultimately, any way of approaching the teaching of piano that implements the core principles discussed in my other posts and makes sense to the student will probably work great.
Position. They should be sitting far enough away that their elbows aren't stuck in their sides, which will limit movement when they start playing songs that use more of the keyboard. And the bench should be low enough that their elbows are a little lower than the level of the keyboard. Most kids end up sitting too high. And they should sit up straight with good posture. Really, the key aspect of this is comfort combined with an active position so they're ready to exert some energy practicing. Teach them to get in the right position very first every time they sit down to play the piano.
Technique. There are layers and layers of technique that become more important when they are playing higher-level music. For example, should you play with fingers curved or fingers flat? Well, it depends on the type of playing you're doing. Students can start to learn those different techniques when the level of music they're playing starts to require them (not until they're playing at least at an intermediate level I'd say). And, at that time, they can read whole books about it. A good starting reference book is Fundamentals of Piano Practice by Chang. It's a free PDF if you Google it. Forewarning, not everything he says in it is true, so it shouldn't be the only reference. Having said all of that, there are two super important playing techniques students should learn from the start because these will apply to all piano playing they do. First, they need to keep all their muscles (finger muscles, hand muscles, wrist muscles, arm muscles, . . .) as relaxed as possible. Stress in muscles will worsen their playing, and it's never necessary, even when they're playing loud or fast. Second, they need to avoid all unnecessary hand movements. Economy of motion will enable them to play cleaner all the time.
Intro to the keyboard. They first need to gain a good understanding of the structure of the keyboard (organized into octaves, find notes based on where they are in relation to the black key groupings, "twins" and "triplets") and be able to find any note by feel (no looking!) with either hand. I purposely teach the notes to them in random order so they don't ever rely on counting up from another note they already know. They should be able to immediately find a note purely based on their memorization of where it is in relation to the black keys. This is their foundation. I also teach them right then what a sharp and flat are (semitone up/down from the index key, which means some sharps/flats are white keys!) so they know all the notes on the entire keyboard, and I'll test them with those too. From there, it's nice to give them a few reference points, so I teach them to find Middle C, the G on the lowest line of the bass clef, and the F on the highest line of the treble clef.
Flashcards. Once they are solid on their intro to the keyboard knowledge, they need to start doing note flashcards each day. I take the treble clef and the bass clef separately so they can get a good idea of the range/location on the keyboard of where the bass clef notes are and where the treble clef notes are. I have them name the note as soon as they see it, and then next they find the note (without looking, of course) and play it. And, importantly, they should be doing these in random order so that they can look at a specific line or space on either of the clefs and know exactly what note it is without having to work their way up/down to it from another note they know better. This is their foundation of fluent music reading. Kids usually don't know how to do flashcards effectively (like knowing to just focus on a few at a time, knowing when they should move one into the "learned" stack, and that they should put one they got wrong at the back of the day's stack so they see it again a minute later), so you'll have to teach them. They will also need to start doing key signature flashcards soon too. I don't know the exact right time, but soon. Notes flashcards and key signature flashcards are the only flashcards I have them do, so they don't have to do any more once they've mastered those.
Hint about notes flashcards: Usually they come with the treble and bass clefs right there beside the note, but I find that students will integrate those clefs into their memorization of where the note is, which means they can't figure out which note they're looking at in actual music when it's not at the beginning of a line. So I used our paper cutter to cut off the left side of all our flashcards so they'll only be looking at the actual location on the grand staff to remember which notes are where.
WEEKLY LESSON STRUCTURE
1. Look at their practice log: This is the very first thing I always do so I don't forget. Make your expectation clear that they should practice six times in between lessons. This is what they need to strive for. But balance that clear expectation with love and understanding when life happens and they don't get all six practice sessions in some weeks. If necessary, this is the time I'll help troubleshoot what got in the way of practicing so they can do better next week.
2. Ear training: Before they've heard a single note played, I will test their absolute pitch by playing a B flat, B, or C and having them guess which note it is. Then I'll play that note in different octaves and have them try to lock the particular color of that note into their brain. (FYI, this isn't the only activity I do to help them develop absolute pitch, which is good since it's probably not enough on its own.) Next I play a single note or a chord and have them sing the note(s) and then find the note(s) on the keyboard. Next I'll test their interval recognition skills by playing either a solid or broken interval. This all takes only a minute, and it's really fun for them.
3. Performance: Next, I want them to get the performance aspect out of the way. This is the part they usually stress the most about, but it's an important part of accountability because they will be more motivated to work hard practicing that week if they know they'll be required to perform the songs for me at their next lesson. I will randomly choose some songs (all of them if we have time) for them to perform. And since they feel pressured with me just sitting there waiting for them to start, I remind them that it's okay and necessary to take some time to do their pre-read routine and mental play it first. The other reason I do this early on in the lesson is because, if their performance ends up being a traumatic experience for them because they didn't practice well, they'll hopefully forget those feelings and be having fun again way before the lesson ends. I try really hard to be understanding and encouraging if they play horribly and, again, troubleshoot if necessary. And if they perform awesomely, I'll let them know it!
4. Show them what they'll practice the next week: We then look through the next week's material and I make sure it's clear to them what they need to be doing. If there are new principles or theory that they might not understand just from reading their lesson books, I'll do a little explanation. The number one goal here is for them have no ambiguity about what they're practicing and learning the next week, otherwise they're unlikely to learn anything. If, on the rare occasion, they showed that they didn't learn anything from last week's material, I will consider having them re-do that material for one more week, but I make sure I understand exactly what went wrong. Did they just not practice enough times? Did I not explain everything well enough? When considering whether to have them re-do a week, keep in mind the importance of continuing to push them forward through the material. They'll pick things up along the way as they see them again and again.
5. Extras: Some of my favorite lessons as a kid were the ones where my teacher was so excited about some music thing that he took time out of our lesson to show me--playing me a song he just found, showing the application of a principle I'd just learned, etc. That enthusiasm often made me think, as I was walking home from my lesson, "Man, I'm just so excited again about music!" Do these things as they strike you, and they may do more to motivate your students than anything else.
These lessons often go past 30 minutes, and maybe that just because I'm slow. But I keep them as short as possible so they're not super fatigued by the end and no longer focusing. I want to end the lesson before they want it to end, which will keep them coming back for more. And once I release them, I'll take a few minutes to write down in their book any instructions I feel they'll need so they can refer to that every day when they practice. For some reason they feel like it's a special surprise when they open up their practice book the next day and see what I've written. (Maybe I should start adding drawings or stickers or something to enhance that, I don't know.) What do I usually write? Usually it's just a list of the things they need to do during their practice session each day: "1. Flashcards. 2. Lesson books pages 18-19." And so on. I always add reminders as well about principles they need to focus on that week, such as to make everything they play beautiful, not look at the keyboard, remember their pre-read and post-read routines, keep the rhythm, etc.
As a parent teaching my children, it's very tempting to do lessons at different times each week, whenever it happens to be convenient. But that will almost certainly mean they don't get their lessons as regularly as they should. So we've found that it's important to have an assigned lesson day and approximate time each week on our calendar (e.g., every Wednesday after dinner).
DAILY PRACTICE ROUTINE
I have them do their flashcards at the beginning and the end of their practice sessions. And they work through their lesson books. Practicing piano using the principles I write about is really hard work, so they probably shouldn't be practicing more than 30 minutes when they're young (maybe only 15 or 20 minutes when they're really young), but I don't set a time limit on their practicing length or have them keep track; I just make sure I don't assign too much for each day and make sure they know how to practice efficiently.
As homeschoolers, we have them integrate their piano practicing into their school day, which makes them way more likely to practice, and it also allows them to do it at the best time for them each day when they have the energy they need to dedicate to it.
We usually only do piano lessons during school times. Vacation times are a good excuse to give them a break from piano so they don't get too fatigued from doing it week after week ad nauseam. In general, the goal is to finish one level each year, so sometimes this means we'll do piano during some non-school weeks if it means they can finish off the level before stopping for a while.
Mixed in with the regular weeks of regular lesson material, I will do an occasional week of fun ear training exercises. And I'll also have them, about once a year, stop their lesson books for a few weeks so they can learn two recital pieces. I let them choose whichever ones are at the right level of difficulty and that they will really enjoy hearing themselves play. After they have learned those (or when they're almost done), I start them up again doing their regular lessons. These breaks from the usual routine are refreshing and motivating for them, so I believe they end up learning better and faster because of them.
LEARNING RECITAL PIECES
Most people think memorizing a recital song is best done by playing it over and over and over until it's finally learned, and then playing it over and over some more until it's finally memorized. This is a recipe for them hating the song before they even get to perform it! And it's ineffective. Here's how I have my kids learn recital songs:
Step 1: Analyze the song and divide it up. Analyzing the song will help them recognize the overall structure of it and especially pick out the repeats. This makes it less daunting because they can see that even though, for example, it's 100 measures long, they only have to learn 40 or so. Next, they are ready to divide the song up into short sections for learning. These sections should be overlapping by a measure or two to allow for smooth transitions between sections.
Step 2: Start memorizing first. Have them choose the most difficult section of the song and memorize it before they ever play it. Then have them practice playing it from memory. They'll have to refer to the music often to make sure they've memorized it properly. And if it's initially too hard to memorize and play hands together, they can do hands separate and then, when each hand is solid enough that they can play even faster than performance speed, they can put them together.
Step 3: Do the other sections. Then they choose the next most difficult section and do the same thing. And proceed like that until they've memorized and can play all the sections. They are purposefully done out of order like this so it will be like a prize at the end when they're finally able to start putting together the sections to make up one amazing beautiful song. Practicing the whole thing really slowly is a good way of testing whether they really have it memorized (as opposed to just relying on finger memory to know which notes come next). They also should be able to mental play the whole song since they've learned it like this.
This process is so much more efficient because they're always playing the song from memory, and it's also more enjoyable for them because they aren't tired of the song before they even finish learning it.
There are also performance techniques to teach them, but one of the best things is to get them to do little performances for family and friends before the big recital. Performing in various settings also helps cement their memory of the song in a way that is separated from their usual environment where they learned it. And as long as they understand that they just need to do their best and have fun, they can relax about how many mistakes they make or how anxious they feel since they can't directly control those things. When they're more experienced, the overwhelming desire to show off/share a beautiful emotional song with the audience should silence most of their performance anxiety.