To Allah belongs the dominion of the heavens and the earth. He creates what He wills (and plans). He bestows (children) male or female according to His Will (and Plan), (Quran 42:49)

The first sound and words a Muslim baby should hear is the call to prayer. The father whispers into his/her child’s pure little ears. The call to prayer is also known as Adhan which Muslims hear five times a day before they face their Lord.

Narrated 'Ubaidullah bin Abi Rafi':
That his father said: "I saw the Messenger of Allah peace be upon him say the Adhan in the ear of AlHasan bin 'Ali - when he was born to Fatimah - the Adhan of Salat."
Jami` at-Tirmidhi, Book of Sacrifices, Hadith 1514

After whispering the call to prayer in the baby’s ears, it is recommended to do Tahneek for the child, accompanied by a supplication. Tahneek means to chew or soften a date and then rub the palate of the newborn child with it. Naming the baby can be done on the day he or she is born or on the seventh day based on the Hadith (narration of the Prophet peace be upon him) below.

Narrated 'Amr bin Shu'aib:
From his father, from his grandfather that the Prophet (peace be upon him) ordered naming the child on the seventh day, removing the harm from him, and Al-'Aqq (removing the hair and slaughtering the animal for 'Aqiqah).
Sahih, authentic
Jami’ At-Tirmidhi, Book of Manners Hadith 2832

It is not permissible to name a child with attributes that are only for Allah such as Ar-Rahman (The All Compassionate), Al-Mutakabbir (The Greatest/ Most Majestic) and so on. It can, however, precede with an “Abd” to indicate the servant’s position of the child before God such as Abd Ar-Rahman (the servant of the Most Merciful). Similarly, the child must not be named as servants of anything other than Allah such as Abdur-Rasul (the servant of Messenger of God) as that would contradict the true meaning of Islam – to submit to One and Only God. In any case, names must not contain any mockery or anything that would cause embarrassment and insult to the child.

On the seventh day after the Muslim baby is born, it is recommended to sacrifice a livestock animal. This is known as Aqiqah. On slaughtering the animal, it is obligatory to mention the name of Allah with the intention the Aqiqah is made for the new Muslim baby. On the same day of the Aqiqah after the sacrifice, the baby’s head can be shaved completely. The weight of the baby’s hair is measured. Then, the parents donate that amount in silver to charity. Regarding circumcision of boys there is another article discussing it in detail.

Muslim women are encouraged to give their children the nourishment of breast milk. The Quran instructs that if a woman breastfeeds her children, their period of weaning is two years. It is common for close family members to visit right away, and for other visitors to wait until a week or more after birth in order to protect the child from exposure to illnesses. The new mother is in convalescence for a period of 40 days, during which friends and relatives will often provide the family with meals.