Prerequisite: student must be comfortable walking around on the horse without holding on to the saddle. Neither legs nor upper body may swing wildly about at the walk, all in all the student should have a decent basic balanced position.
Step one: figure out how bold the student is by leading the horse for a few steps of sitting trot. If student grins, giggles, asks to do it again, etc, you're dealing with a student of the bold variety. If the student is scared, I will tell them that they don't need to trot now and to tell me when they're feeling ready. We'll do all the walking exercises and practice our mounting/dismounting and steering until the student feels confident enough to trot.
Step two: at the halt, coach student through vertical far (standing totally straight up in the stirrups) and halfseat positions. Student needs to the move from fullseat to vertical far and back again smoothly and without their hands on the saddle for balance.
Step three: Repeat step two at the walk - the student needs to be able to 'post' with basically good mechanics. This should vary between holding vertical far for ten seconds at a time and being able to post on the correct "diagonal" at the walk. Without their hands on the saddle for balance, the lower leg must remain still before moving on to the trot.
Step four: Remind the rider to stand up when you say up and sit gently down when you say down, and then spend a few circles counting off, "up, down, up, down" etc. Don't try to engage too much from the kid - allow them to just feel the trot rhythm and listen to you. They are allowed to hold onto the saddle. Then come to the walk and ask if they could feel the rhythm that you were counting out and if they're ready to work towards posting.
This bit often takes a while. Some variations on this that I've seen include the student bouncing wildly and insisting they're posting (more common than you'd think), sitting/standing too long, losing their lower leg in nearly any direction, fetal crouch coming on suddenly, and becoming afraid again. Guide them through the corrections by coming to the walk/halt if necessary and making sure that they completely understand what you want. (e.g. draw your leg back underneath you... what on Earth does that mean?! or how about heels down? Heels down where? Does the lower leg go forward? How do I know when my heels are down?)
Once the rhythm seems to be established and they're mostly going up and down with the horse....
Step five: Talk to them. If the student can't hold the rhythm and answer questions about their favorite ice cream, meal, movie, color, pet, sibling, etc. then the tempo isn't really in their muscles yet and it's taking up too much of their immediate focus to move on.
Step six: Start an independent upper body by asking them to lift one hand up above their head, then switch. Once they can switch smoothly without loss of posting, I start encouraging them to let go with both hands.
Step seven: Move their arms through a variety of positions. I like: hands totally into the clouds, hands on your head, hands on your hips like you're mad at me, stretching forward as if to touch an ear, leaning back as if to touch the tail, hands behind the back. If the lower leg stays relatively stable during all of this, they're about ready for the reins.
Step eight: Now lots of walk-trot trot-walk transitions. Beginning the posting trot without reverting to holding onto the saddle and without falling forward or backwards is crucial to allowing the rider to post alone.
Step nine: At the halt, show the rider how their elbows open and close during the action of posting trot, where their hands should be, and why. Have them practice keeping their hands still while posting at the walk.
Step ten: Allow the rider to get the horse to trot on their own, as well as ask the horse to walk again, keeping them on the longe for extra control for a while.
Rider coaches out there, do you have anything to add? Do you teach it differently or think I'm missing anything important?