What are some practical lessons on horse care that are provided as well as teachings that still apply today in Black Beauty?

Horses are sensitive and in some ways fairly delicate animals; whoever cares for them must have knowledge that they are different from other common domestic animals.
Anna Sewell said herself that she wrote the tale of Black Beauty "to induce kindness, sympathy, and an understanding treatment of horses" Sewell's intention in writing Black Beauty was not primarily to write a children's novel, but to inform readers specifically about the evils of the bearing rein. In addition to this issue, Sewell hoped to teach others about the care and feeding of horses, and, especially, the humane treatment of horses since they were the primary means of transportation in her time.
Some lessons:
Because horses are prey animals (as opposed to predators), the first instinct is to run and flee if anything frightens them or is perceived as a danger. Therefore, owners must understand why they react to some foreign sound or sight and not punish them, but teach them, instead.
Being among this group of prey animals, horses must be conditioned to ignore loud noises or things approaching them such as other carts or carriages.
Teaching a horse to pull a carriage is easier if it is paired with another who is already trained for this task.
Because horses are herd animals, they will sometimes learn bad habits from other horses if they are around those who misbehave.
When a horse has been run to the point that it is very heated, it must be walked around and allowed to cool down. Never should cold water be given to a hot horse or fed until it cools down. 
A mean or hard owner will "wear all the spirit out of [a horse] and just make [it] into a quiet, humble, obedient piece of horseflesh." (These are Ginger's words.)
Having a farrier who is competent is essential since the health of a horse's foot is paramount.
When Ginger tells her life history to Beauty, she mentions that "fashion is one of the wickedest things in the world." Specifically, she tells Beauty of the cruel bearing rein that forced carriage horses to hold up their heads at all times, even if going uphill. Also, many owners would cut the tails off the horses, when tails are essential for whisking flies away from their flanks and underside.
Some teachings on horses:
Although horses are not loving like dogs, they do demonstrate some affection and caring for kind owners.
Horses have feelings and high intelligence. Mistreatment of a horse is not only cruel, but foolhardy if one wishes to have a horse with some spirit. For, a horse who is beaten will act unthinkingly, and if broken in spirit, a horse will not live as long as if it were loved.
Horses definitely have a strong sixth sense as Beauty demonstrates when she refuses to cross the bridge that is unsafe. They somehow can sense where a hidden ditch is or some difference in the terrain that could be dangerous. Then, they will balk and not go forward as Beauty does on the bridge because it is dilapidated.
Horses have an incredible sense of direction and can find their way home.
Just like other higher-level animals, horses need a clean, bright environment. They need to be fed on a regular basis and have clean water. They need fresh air and exercise, but not too much work, and they need brushing and hoof-cleaning and other acts of attention. As a herd animal, they like company.
People who think horses are stupid are only ignorant of the nature of horses. Often horses will balk or misbehave because they know that the human is ignorant of their ways, or they have picked up such habits from other horses.