Generally, the school system appears to try to teach children what they need to know for their age group, based on what developmental psychologists indicate children of that age should be able to grasp. There is no point in trying to teach a child something that the child is not yet old enough to comprehend, and there is also no point in waiting until a child is already involved with something before teaching him or her about it, such as teaching children about substance abuse and sexual activity, which are both serious issues in schools today. Learning itself generally defies description, as there are so many ways to define it. However, I think that learning is the acquiring of knowledge that one did not have before, that is important for any reason, and that will benefit someone during their lifetime. It does not really matter how that knowledge is acquired, overall, as long as the knowledge is 'learned' before the issue learned about harms the one that is learning. Learning comes from a combination of emotional effects, associations that a child makes between one thing and something else, the conditioning that an individual goes through, and the various cognitive processes that take place as a child is learning and growing.

As for who should be responsible in the teaching and learning of children, this depends on the age of the child. When children are in infancy, the parents and family are generally responsible...

As they get a little bit older, children attend Pre-K programs, Kindergarten, and then elementary school. While they are in the school environment, the teachers and other staff members are responsible for the learning that takes place, but the parents are still responsible at home. The same is generally true throughout the school careers of these children. When they got to early adulthood, however, they are out of school, and unless they attend college, they are much more responsible for what they learn than others are. In other words, they are 'grown up' at that point, and they are able to do as they please.
This does not mean that they will not continue to learn from others, but only that they are generally no longer obligated to learn from others. They move into the world, learn their own lessons about life, love, money, and all of the others issues that make up their existence, and they take those lessons and (hopefully) use them to either avoid making mistakes or not make the same mistakes that they have made in the past. This does not always work well, but a child that has had a good upbringing will generally avoid making too many serious mistakes, and the parents that have taken care to teach that child all that they can about life and society will often not worry quite as much about that child as he or she grows into an adult and begins to make his or her own way in the world.