When I was riding, there were four basic speeds: Walking, Trotting, Cantering, Galloping. Walking, is self explanatory. Trotting is relatively slow and bumpy as fuck so you typically have to "post" [essentially rapidly change from standing up and sitting down]. Cantering is significantly smoother [you just sit] and faster, and galloping is like a faster canter. Before we did anything faster than a trot, we typically warmed up the horses for a few minutes.
For jumping, we typically changed our "stance" to something called a two point one where we kindof stand, leaning towards the horse's neck with our hands up, although I'm not sure if it was significant. There's also something called the three point stance [or something...] that you use for higher jumps and you're higher up on the horse.
I was never too good at horseback riding [I was a wee little kid at the time] and thus I had to use reins in order to direct my horse. My instructor said that you only need to use your legs [as in pressuring one side would change the horses direction] and did that while riding bareback.
The way I was taught to change speeds [I'm assuming that it's based off of how the horse was broken], was that a kick with both legs is used to trot. Exerting pressure with the left leg and kicking three times during a trot was used for cantering, and kicking or pressuring with both legs during a canter was used to gallop.
As for horsebumping, when I was riding if my horse ran or walked into one of the posts we had in the ground [like a sideways cylindrical piece of wood], I was able to feel it, not sure if it was enough to hurt the horse's ankles, but it wasn't enough to bump me.
Horses scare like fuck. My first day, I was outside and there was a plain overhead. My horse started to randomly jump sideways and I was like O.o.
I don't know how to rear a horse on command
[which should be a mappable button imo].
Spurs and crops are not going to damage a horse.
The last thing I remember is that it's actually a lot easier to learn than I thought it would be.