The Torah tells us that Hashem loves Avraham Avinu, and the reason is that he teaches his children to go in Hashem's ways. As it states (Bereishis 18:19) אֶת יְצ ַוֶּה ֶׁראֲש לְמַעַן יְדַעְת ִּיו כִּי וּמִשְׁפ ָּט צְדָקָה לַעֲש ׂוֹת 'ה דֶּרֶ ךְ ָׁמ ְ רוּוְש אַחֲר ָיו בֵּיתוֹ וְאֶת בָּנָיו , "For I have loved him, because he commands his children and his household after him that they keep the way of Hashem, doing tzedakah and mishpat."
This is what Hashem wants from every parent: to teach their children to follow the ways of the Torah.
The Ramban (Devarim 6:7) writes pesukim in the Torah that show how Hashem wants the religion to be passed down from parent to child, throughout all generations. The Ramban writes, "The Torah commands us (Shemos 31:17) לעולם היא אות ישראל בני ובין ביני, 'Between Me and Bnei Yisrael, it is a sign forever...' And it states (Bereishis 17:10) בריתי זאת אחריך זרעך ובין וביניכם ביני תשמרו אשר , 'This is my bris which you shall keep between Me and you and your offspring after you...' But how will our children know the mitzvos if we don't teach them? This is the reason we have a mitzvah (Devarim 6:7) לבניך ושננתם, 'you shall teach them thoroughly to your children,' the mitzvah to teach Torah to our children."
Someone asked the Satmar Rav zt'l regarding the people who were killed during the Holocaust, whether they can be considered as being killed al kidush Hashem. The Satmar Rebbe replied that it is certainly al kidush Hashem. He said, "Those who were killed in the Holocaust are kedoshim, but those who survived the Holocaust and raised a family to serve Hashem are Kodesh Kadoshim."
One aspect of chinuch is to guard one's children from negative influences. Included in this is knowing who the child's friends are, who he plays with. These are important questions to ask and find out, because a friend is a strong influence.
In parashas Yisro, we read that Yisro came to the desert together with his daughter Tzipora (Moshe's wife) and with Moshe's children, Gershom and Eliezer. They came to join the Jewish nation in the desert. There is an opinion that says that Yisro came to the desert only after matan Torah. According to this view, we wonder why Moshe Rabbeinu didn't call his children to come earlier, so they could be present when the Torah was given?
Reb Meir Tzvibel answers that Moshe Rabbeinu didn't want his children to be at Matan Torah without supervision. When he would be on the mountain, receiving the Torah from Hashem, who would watch over them? His children would mingle with the crowd, but with whom? Better for his children to remain in Midyan with his wife Tzipora watching over them, than to be at matan Torah, on their own.
What greater merit could there be than to be present at Matan Torah?! Chazal tell us that at this time, the nation became entirely pure from their tumah, זהומתן פסקה. Nevertheless, Moshe felt it wasn't worthwhile, if that meant that his children wouldn't be supervised. Kal v'chomer that every parent should seek to know where their children are, with whom they play, etc., because chinuch habanim is so essential.
It states (Devarim 4:10) יְלַמ ֵּדוּן בְּנֵיהֶם וְאֶת "They shall teach their children," and afterwards, the next pasuk states, ָּׁמ ַיִםהַש לֵב עַד ׁבָּאֵש בֹּעֵר וְהָהָר, "the mountain was burning with fire up to the heart of heaven." This tells us that the way to educate children is when the parent (or the teacher) has a fire in his heart, burning עַד ָׁמ ַיִםהַש לֵב , up to the heart of heaven.
It states (Devarim 4:9) מ ְאֹד ָנַפְשְׁך וּש ְׁמֹר ָלְך ָּׁמֶרהִש רַק יְמֵי כֹּל ָמ ִלְּב ָבְך יָסוּרוּ וּפֶן עֵינֶיךָ ר ָאוּ ֶׁראֲש ה ַדְּב ָרִים אֶת ת ִּש ְׁכַּח פֶּן בָנֶיךָ וְלִבְנֵי לְב ָנֶיךָ וְהוֹדַעְת ָּם חַיֶּיךָ , "Only be aware for yourself, and greatly be aware for your soul, lest you forget the things your eyes have beheld and lest you remove them from your heart all the days of your life, and make them known to your children and your children's children."
The Tiferes Shlomo asks, if one forgot, how can he teach his children? The pasuk is referring to those who don't daven with kavanah; they just say the words, and they are lax with their avodas Hashem, but they rebuke their children and tell them to go on the right path. How can a parent give mussar if he isn't cautious himself? This is what the pasuk is referring to when it states, יָסוּרוּ וּפֶן בָנֶיךָ וְלִבְנֵי לְב ָנֶיךָ וְהוֹדַעְת ָּם ...ָמ ִלְּב ָבְך . If the parents are lax, how can they teach their children? Rather the solution is as it states in the next pasuk, הֵם ֶׁראֲש ה ַיָּמ ִים כָּל אֹתִי לְיִרְאָה יִלְמ ְדוּן ֶׁראֲש יְלַמ ֵּדוּן בְּנֵיהֶם וְאֶת הָאֲדָמָה עַל חַיִּים , "so that they shall learn to fear Me all the days that they live on the earth, and they shall teach their children." First, the parents have to fear Hashem, and then they will be able to teach and influence their children.