What THREE things
should every child be taught?

1. To be obedient to parents.

2. About Jesus the Messiah and the GOOD NEWS of God's kingdom.

3. The laws (commandments) of God.

1) Children should be taught to be obedient to parents:

“Children, obey your parents in the Lord, for this is right. Honor your father and mother (which is the first commandment with a promise), so that it may be well with you, and that you may live long on the earth." (Ephesians 6:1-3, NASB throughout unless noted)

“Children, obey your parents in all things: for this is well pleasing unto the Lord.” (Colossians 3:20)

" . . . For what son is not disciplined by his father?  If you are not disciplined (and everyone undergoes discipline), then you are illegitimate children and not true sons.  Moreover, we have all had human fathers who disciplined us and we respected them for it.  How much more should we submit to the Father of our spirits and live!  Our fathers disciplined us for a little while as they thought best; but God disciplines us for our good, that we may share in his holiness.

"No discipline seems pleasant at the time, but painful.  Later on, however, it produces a harvest of righteousness and peace for those who have been trained by it." (Hebrews 12:7-11, NIV)

"For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, . . .

"And even as they did not like to retain God in their knowledge, God gave them over to a debased mind, to do those things which are not fitting . . . backbiters, haters of God, violent, proud, boasters, inventors of evil things, disobedient to parents . . ." (Romans 1:18, 28, 30, NKJV)

"But know this, that in the last days perilous times will come: For men will be lovers of themselves, lovers of money, boasters, proud, blasphemers, disobedient to parents, unthankful, unholy . . . having a form of godliness but denying its power. And from such people turn away!" (2Timothy 3:1-3, NKJV)

2) Children should be taught about Jesus the Messiah:

"Then Jesus called a little child to Him, set him in the midst of them, and said, 'Assuredly, I say to you, unless you are converted and become as little children, you will by no means enter the kingdom of heaven. . . . Whoever receives one little child like this in My name receives Me. . . .

“Whoever causes one of these little ones who believe in Me to sin, it would be better for him if a millstone were hung around his neck, and he were drowned in the depth of the sea. . . . Take heed that you do not despise one of these little ones, for I say to you that in heaven their angels always see the face of My Father who is in heaven.' " (Matthew 18:2-3, 5-6, 10, NKJV)

"Then they brought little children to Him, that He might touch them; but the disciples rebuked those who brought them.

"But when Jesus saw it, He was greatly displeased and said to them, 'Let the little children come to Me, and do not forbid them; for of such is the kingdom of God. Assuredly, I say to you, whoever does not receive the kingdom of God as a little child will by no means enter it.' And He took them up in His arms, laid His hands on them, and blessed them." (Mark 10:13-16, NKJV)

 3) Children should be taught about the laws (commandments) of God:

“Children, obey your parents in the Lord, for this is right. Honor your father and mother (which is the first commandment with a promise), so that it may be well with you, and that you may live long on the earth." (Ephesians 6:1-3)

"His (Jesus') parents went to Jerusalem every year at the Feast of the Passover. And when He was twelve years old, they went up to Jerusalem according to the custom of the feast. When they had finished the days, as they returned, the Boy Jesus lingered behind in Jerusalem.

"Now so it was that after three days they found Him in the temple, sitting in the midst of the teachers, both listening to them and asking them questions." (This verse also indicates how the children were taught; they were taught by discussions about God's law with adults in the temple.) (Luke 2:41-43, 46, NKJV)

“But continue thou in the things which thou hast learned and hast been assured of, knowing of whom thou hast learned them; And that from a child thou hast known the holy scriptures, which are able to make thee wise unto salvation through faith which is in Christ Jesus. ”  (2Timothy 3:14-15)

Your question specifically asked for scriptures from the New Testament, however, the Bible is intended to be congruous from Genesis to Revelation. In fact, the writers of the New Testament only had what we call the Old Testament to read from. One of the many places in the Old Testament that discuss principles for bringing up and teaching children can be found in the book of Proverbs.