From the time the Law of Moses was given, the Jewish people have been teaching that law to both their children and to each other as adults. This includes both the Bible that we would recognize, and a large body of oral law that was given at the same time as the written law according to Jewish tradition. They use this oral law to help interpret the written law. This oral law was not written down until much later. Both were taught to Jewish men, and are now also taught to women.
In Christ's time, the Sadducees and Pharisees and other rabbinical groups were teachers. The priests were responsible for what went on in the temple - the sacrifices, feasts, and similar functions. These other groups were responsible for instruction and this teaching occurred in both the temple area and in specific houses of study in the Jewish community. These institutions are not described particularly in the Bible, but their teaching members certainly were - the temples themselves get the most press. Clearly by Jesus Christ's time, His view of many of these teachers was not favorable. It is to their discredit that they did not recognize Christ for who He was in spite of all their formal study of the oral law, but that does not discount that such study did occur.
Clearly, this is a challenge to all who teach today as well. We must be sure to impart both the truth of the Bible and also the Spirit of the Bible so that all may find Jesus Christ as their Savior and not be lost. For this, God designated some to be teachers, even in the early church. He also imparted gifts of knowledge and wisdom through the workings of the gifts of the Spirit which can be immensely profitable for the church. But this does not eliminate His teaching ministries. And for the modern man, a web search engine on your smart phone doesn't either.
This is not to slight the contribution of parents or ministers in teaching either. A parent will do more to shape his or her children for right or for wrong than any other individuals, and this is how it should be. You do have them for much of their early years and it is your responsibility to teach and instruct them in what is right and what is wrong, along with all of the other things you must teach them (alphabet, numbers, reading, arithmetic, science and anything else they struggle with in school until they hopefully pass you in learning). But there is value in hearing the view of the Word of God from many individuals beyond a parent and a pastor. If your children only have a couple of viewpoints and haven't had to actually think about what they believe, and defend what they believe, your kids will be easy prey when they go into the real world which the vast majority of them will do.
It is said that two months each year when they were not needed in the fields to work, the Jewish men would go to study the word from these teachers. We would be hard pressed to take two months of every year and dedicate it to the study of God's word today. But we should certainly be able to come a bit early on Sunday mornings to attend Sunday School.