> That got me interested: since the wiring is so long (from limbs to cerebellum), what kinds of motor learning?

I'd imagine it's things like training a dominant hand. The skills required for precise motor control, to produce the right movements for e.g. handwriting. Since the wiring is so long, and feedback is delayed, you need to be able to precalculate these movements.

Also imagine how e.g. an intent to move somewhere actually gets implemented. You don't always have to think about each individual step of walking, or pay explicit attention to things like your sense of balance. You probably don't even have to choose that you're going to walk, or think about how to get up. When you want to go somewhere, you just do it, and somehow it's all calculated for you and happens.

When you try to move a specific limb, how do you know which muscles correspond to that limb? In fact, how many of those muscles can you even individually address? You can learn to individually address them, but I bet you don't come with that ability by default.

Then of course there's the question of what even causes your limbs to move once you will them to move.

What do you mean? The cerebellum is closer to the spinal cord than the rest of the brains. And there's no learning happening anywhere but the brain, vertebrates don't have distributed central nervous system like octopuses do. The only thing vertebrate limbs can do on their own are certain hardcoded reflex actions.

So the spinal cord is only a wire and, besides reflexes, there is no processing done anywhere else, but the brain?

How about neurons in the gut, I've read somewhere that intestines are lined up with some neural "mesh". Is that only for signal transmission?

There is processing in the autonomous nervous system, all sorts of regulation, and adaptation, perhaps something that could be called learning – we don’t really know what happens in the gut nerve complex yet, as it’s so recent "discovery".

Fully autonomous muscle movements like heartbeat, or the peristaltic motion of the gut, are not directly controlled by the brain, but they certainly respond to signals sent by the brain – eg. brain processes sensory data, determines there’s a danger, upregulates the pituitary gland which starts secreting adrenocorticotrophic hormone, which once it reaches your adrenal glands, causes them to start producing adrenaline, which in turn causes all sorts of changes all over the body related to the fight-or-flee reaction.

The skeletal muscle, though – I don’t think its innervation is capable of anything but reflexive motion without the brain telling it what to do.