Children are special people. They have their own identity. Our beloved Prophet Muhammad ﷺshowed this through his behavior. He loved children and was always compassionate and merciful towards them. He showed his affection for them in many ways. He hugged them and patted them on the back. He touched their heads, combing their hair with his fingers. The Prophet Muhammad ﷺloved to play with children. 
Abu Huraira, narrated that the Prophetﷺsaid, "Every child is born with a true faith (i.e. to worship none but Allah Alone) but his parents convert him to Judaism or to Christianity or to Magainism, as an animal delivers a perfect baby animal. Do you find it mutilated?" Then Abu Huraira recited the ayath: "So set you (O Muhammad) your face towards the religion of pure Islâmic Monotheism Hanifa (worship none but Allâh Alone) Allâh's Fitrah (i.e. Allâh's Islâmic Monotheism), with which He has created mankind. No change let there be in Khalqillâh (i.e. the Religion of Allâh Islâmic Monotheism), that is the straight religion, but most of men know not. [Tafsir AtTabarî, Vol 21, Page 41] " (30.30). Sahih Al Bukhari 1358 (Vol.2:440)
The Messenger of Allaah ﷺsaid, according to a hadeeth (prophetic narration) narrated by ‘Abd-Allaah ibn ‘Umar, “… and your child has rights over you.” Sahih Muslim, 1159
The Prophet ﷺ used to seek Allah's protection for Al-Hasan and Al-Husain by saying: (Dua to safeguard from Evil Eyes) "U'eezukumaa bikalimaatil-laahit-taammati min kulli shaytaanin wa haammatin, wa min kulli 'aynin laammatin".
Translation: "I seek protection for you in the Perfect Words of Allah from every devil and every beast and from every envious blameworthy eye" Sahih Al Bukhari Vol 4:119.
It was narrated from ‘Abd-Allaah (may Allaah be pleased with him) that the Messenger of Allaah ﷺsaid: “Each of you is a shepherd and is responsible for his flock. The ruler who is in charge of people is a shepherd and is responsible for them. The man is the shepherd of his household and is responsible for them. The woman is the shepherd of her husband’s house and child and is responsible for them. The slave is the shepherd of his master’s wealth and is responsible for it. Each of you is a shepherd and each of you is responsible for his flock.” Sahih al-Bukhari, 2416; Muslim, 1829
So parents must take care of teaching their children the duties of Islam and other virtues that are recommended in sharee’ah, and worldly matters that they need in order to live a decent life in this world.
The man should start by teaching them the most important things, then the next most important. So he starts by teaching them correct ‘aqeedah (creed), free from shirk (polytheism)and bid’ah (innovation). Then he teaches them the acts of worship, especially prayer. Then he teaches them and trains them in good manners and characteristics, and everything that is good.
Allaah says (interpretation of the meaning): “And (remember) when Luqmaan said to his son when he was advising him: “O my son! Join not in worship others with Allaah. Verily, joining others in worship with Allaah is a great Zulm (wrong) indeed” Quran Surah Luqmaan 31:13
It was narrated from ‘Abd al-Malik ibn al-Rabee’ ibn Sabrah from his father that his grandfather said: The Messenger of Allaah ﷺsaid: “Teach the child to pray when he is seven years old, and smack him (lightly)if he does not pray when he is ten.” Tirmidhi, 407; Abu Dawood, 494. Classed as saheeh by Shaykh al-Albaani in Saheeh al-Jaami’, 4025
Every father and mother should train their children in praiseworthy characteristics and good manners, whether towards Allaah, His Prophet the Messenger of Allaah ﷺ, towards thei Qur’aan and their ummah (global community of believers), and with everyone whom they know and who has rights over them. They should not behave badly with those whom they mix with, their neighbours or their friends.
The father must discipline his child and teach him what he needs to know of religious duties. This teaching is obligatory upon the father and all those in charge of children before the child reaches the age of adolescence. This was stated by al-Shaafa’i and his companions. Al-Shaafa’i and his companions said: This teaching is also obligatory upon the mother, if there is no father, because it is part of the child’s upbringing and they have a share of that and the wages for this teaching may be taken from the child’s own wealth. If the child has no wealth then the one who is obliged to spend on him may spend on his education, because it is one of the things that he needs. And Allaah knows best. Sharh al-Nawawi ‘ala Saheeh Muslim, 8/44
He must also protect his sons and daughters from everything that may bring them close to the Fire. Allaah says: “O you who believe! Ward off yourselves and your families against a Fire (Hell) whose fuel is men and stones, over which are (appointed) angels stern (and) severe, who disobey not, (from executing) the Commands they receive from Allaah, but do that which they are commanded” Quran Surah Tahreem 66:6
He used to kiss children and loved them very much. Narrated by Abu Huraira(R) that Allah's Messenger (ﷺ) kissed Al-Hasan bin `Ali while Al-Aqra' bin H`Abis at-Tamim was sitting beside him. Al-Aqra said, "I have ten children and I have never kissed anyone of them," Allah's Messenger (ﷺ) cast a look at him and said, "Whoever is not merciful to others will not be treated mercifully." Sahih al-Bukhari 5997 (Vol. 8: 26)
Narrated Jabir bin AbdullahAllah's Apostle said, "When night falls (or it is evening), keep your children close to you for the devils spread out at that time. But when an hour of the night elapses, you can let them free. Close the doors and mention the Name of Allah, for Shaytaan (Satan) does not open a closed door."Sahih Al Bukhari 4:523
Allah said: Your wealth and your children are only a trial, whereas Allah! With Him is a great reward (Paradise). Quran Surah Taghabun 64:15
Hudhaifa said, "The affliction of a man in his family, his property, his children and his neighbors are expiated by his prayers, Zakat (and alms) and enjoining good and forbidding evil. Sahih al-Bukhari 7096 
This is one of the father’s obligations towards his children; it is not permissible for him to fall short in that or to neglect this matter, rather he is obliged to do this duty in the fullest sense.
It was narrated that ‘Abd-Allaah ibn ‘Amr (may Allaah be pleased with him) said: The Messenger of Allaah ﷺsaid: “It is sufficient sin for a man if he neglects those on whom he is obliged to spend.” Abu Dawood, 1692, classed as sahan by Shaykh al-Albaani in Saheeh al-Jaami’, 4481.
Another of the greatest rights is to give the child a good upbringing and take good care of him or her – especially in the case of girls. The Messenger of Allaah ﷺencouraged this righteous deed.
It was narrated that ‘Aa’ishah the wife of the Prophet (peace and blessings of Allah be upon him) said: A woman came to me with two daughters and asked me for food, and I could not find anything except one date which I gave to her. She shared it between her two daughters, then she got up and went out. The Prophet ﷺcame in and I told him what had happened. He said: “Whoever is in charge of any of these girls and treats them well, they will be a shield for him against the Fire.” Sahih Al Bukhari, 5649; Muslim, 2629
Another important matter which is one of the rights of children to which attention must be paid, is treating children fairly. This right was referred to by the Prophet ﷺin the saheeh hadeeth (authentic narration): “Fear Allaah and treat your children fairly.” (Narrated by al-Bukhaari, 2447; Muslim, 1623). It is not permissible to show preference to females over males, just as it is not permissible to show preference to males over females. If the father makes this mistake and shows preference to some of his children over others, and does not treat them fairly, this will lead to many evils, such as:
The harm that befalls the father himself, for the children whom he denies or deprives will grow up to hate him. The Prophet ﷺreferred to this in the hadeeth narrated by Sahih Muslim 1623 when he said to the father of al-Nu’maan, “Would you like them to honour you equally?” He said, “Yes.” In other words, if you want them all to honour you equally, then be fair in giving gifts to them. 
Prophet Muhammadﷺnever held back his love for the children and always expressed his fondness to them. He would pick up children in his arms, play with them, and kiss them. He played with them and took such keen interest in them.
Narrated Abu Hurairah (may Allah be pleased with him), Allah’s Messengerﷺkissed Al-Hasan ibn `Ali while Al-Aqra` ibn Habis At-Tamim was sitting with him . Al-Aqra` said, “I have ten children and have never kissed one of them.” The Prophet cast a look at him and said, “Whoever is not merciful to others will not be treated mercifully.” Sahih Al-Bukhari 5997 (Vol.8:26)
In one hadith Abu Hurairah (may Allah be pleased with him) narrated: I went along with Allah’s Messengerﷺat a time during the day but he did not talk to me and I did not talk to him until he reached the market of Banu Qainuqa`. He came back to the tent of Fatimah and said, “Is the little chap (meaning Al-Hasan) there?” We were under the impression that his mother had detained him in order to bathe him and dress him and garland him with sweet garland. Not much time had passed that he (Al-Hasan) came running until both of them embraced each other, thereupon Allah’s Messengerﷺsaid, “O Allah, I love him; love him and love one who loves him.” The Virtues Of Al-Hasan And Al-Husain (RA), Sahih Muslim 2421
Anas ibn Malik (may Allah be pleased with him), the servant of the Prophet, had another recollection: I never saw anyone who was more compassionate towards children than Allah’s Messengerﷺ.His son Ibrahim was in the care of a wet nurse in the hills around Madinah. He would go there, and we would go with him, and he would enter the house, pick up his son and kiss him, then come back. Sahih Muslim 2316
Whenever the Prophet Muhammad (صلی اللہ علیہ وسلم ) passed by children, he tried to be the first to greet them and say “Assalaamu Alaikum.” When riding he would let children sit on his camel or donkey. Sahih Muslim 2168
On the authority of Abdullah bin Abbas, we said: One day I was behind the Prophet and he said to me: “Young man, I shall teach you some words [of advice]: Be mindful of Allah, and Allah will protect you. Be mindful of Allah, and you will find Him in front of you. If you ask, ask of Allah; if you seek help, seek help of Allah. Know that if that Nation were to gather together to benefit you with anything, it would benefit you only with something that Allah had already prescribed for you, and that if they gather together to harm you with anything, they would harm you only with something Allah had already prescribed for you. The pens have been lifted and the pages have dried” Jami at Tirmidhi Vol. 4: 2516
Prophet صلی اللہ علیہ وسلم used to teach children that failure doesn’t exist. Narrated Anas (RA): I served The Prophet for ten years, and he never said to me, “Uf” (a minor harsh word denoting impatience) and never blamed me by saying, “Why did you do so or why didn’t you do so?” Kitab Al-Adab, Sahih Al-Bukhari 6038 (Vol.8: 64)
In a similar hadith narrated by Anas RA reported: “I served the Messenger of Allah (ﷺ) for nine years, and I do not know (of any instance) when he said to me: Why you have done this and that, and he never found fault with me in anything.” Sahih Muslim 2309
We must also become a mentor and help the child believe in his or her ability to succeed no matter how long it takes!
Many of the inappropriate behaviors of young children can simply be ignored or disregarded. The Prophet is our best example in this regard.
Anas bin Malik RA said: “The Messenger of Allah had the best disposition among people. One day he sent me on an errand and I said, ‘By Allah, I will not go,’ but it was in my mind that I would do as the Messenger of Allah had ordered me. I went until I came upon children playing in the street. Then the Messenger of Allah arrived and he caught me by the back of my neck from behind. As I looked at him, I found him smiling, and he said, `Unays (nickname of Anas), did you go where I asked you to go’?’ I said, `O Messenger of Allah, yes, I am going.”‘ Sahih Muslim 2310 a, 2309 e