How can I get a horse to speed up or slow down by using my seat?

I’m unsure this is something you really want to condition your horse to do, however, I will give you your answer before trying to talk you out of deploying it too readily:

1. Firstly, your horse must already be responding to something in order to speed up and slow down now, else this a separate issue. Presuming he is, simply …

2. Begin preceding what you do now to slow down with the seat effect you want to use in the future, e.g. – weight to back for slowing. followed by the hand (reins) if that’s your present means

3. Begin preceding what you do now to accelerate with the seat effect you want to use in the future, e.g. – weight forward for acceleration, followed by your heel if that’s your present means

Horses are particularly gifted with unwinding precedents, and anticipating what you’ll want next this time based on what you wanted next last time.

That said, if you are asking the question, you may be unready for this degree of fidelity, as it can have a range of unintended consequences. In all likelihood, as you experience some of them, you will find yourself wanted him to move off your seat only sometimes, but not others – e.g., you lose your balance, and unintentionally shift your weight – simultaneously punching the accelerator (and/or slamming on the brakes), eventually finding yourself correcting him for doing exactly what you just taught him to do.

Horses are particularly un-gifted with vague requirements and contradiction – much like computers. Much like you. And me, of course.

If your horse is properly finished, he should be amply responsive to your hand and heels, such that the use of them is imperceptible, visually – and this is ample fidelity for most purposes, even through high school. Therefore, I might recommend you focus on refining your application of these aids to perfection (remaining free to open and close gates while mounted, swat flies, and so on), and save the aid of your seat and body weight to augment his balance in motion.