The Prophet (peace and blessings of Allah be upon him) said: “There are no days greater in the sight of Allah
and in which righteous deeds are more beloved by Him than these ten days, so during this time recite a great deal
of Tahleel (“La ilaaha ill-Allah”), Takbeer and Tahmeed.” Classed as saheeh by al-Albaani

The virtue of Takbeer
The first ten days of Dhu’l-Hijjah are venerated days. Allah swore by them in His Book, and swearing by a thing is indicative of its importance and great benefit.

Allah says (interpretation of the meaning): “By the dawn; By the ten nights (i.e. the first ten days of the month of Dhul-Hijjah)” [al-Fajr 89:1]

Ibn -Abbaas, Ibn al-Zubayr, Mujaahid, and others among the earlier and later generations said: This refers to the ten days of Dhu’l-Hijjah. Ibn Katheer said: This is the correct view. Tafseer Ibn Katheer, 8/413

Good deeds during these days are beloved by Allah because the Prophet (peace and blessings of Allah be upon him) said: “There are no days in which righteous deeds are more beloved to Allah than these ten days.” They said, “O Messenger of Allah, not even jihad for the sake of Allah?”
The Messenger of Allah (peace and blessings of Allah be upon him) said, “Not even jihad for the sake of Allah, except in the case of a man who went out to fight, giving himself and his wealth up for the cause, and came back with nothing.” Narrated by al-Bukhaari, 969; al-Tirmidhi, 757.

This version was narrated by al-Tirmidhi and classed as Saheeh by al-Albaani in Saheeh al-Tirmidhi, 605.

One of the righteous deeds during these days is to remember Allah by reciting Takbeer (saying “Allahu akbar”) and tahleel (saying “Laa ilaaha ill-Allah”), because of the following evidence.

1 – Allah says (interpretation of the meaning):
“That they may witness things that are of benefit to them (i.e. reward of Hajj in the Hereafter, and also some worldly gain from trade), and mention the Name of Allah on appointed days” [al-Hajj 22:28]

“The appointed days” are the first ten days of Dhu’l-Hijjah.

2 – Allah says (interpretation of the meaning):
“And remember Allah during the appointed Days” [al-Baqarah 2:203]

These are the days of al-Tashreeq, i.e. the 11th, 12th, and 13th of Dhu’l-Hijjah.

3 – The Prophet (peace and blessings of Allah be upon him) said:
“The days of Tashreeq are the days of eating, drinking, and remembering Allah.” Narrated by Muslim, 1141.

Description of the Takbeer
The scholars differed concerning the form of the Takbeer.

1 – The first view is that it is:
Allahu akbar, Allahu akbar, laa ilaaha ill-Allah, Allahu akbar, Allahu akbar, wa lillaahi’l-hamd
(Allah is Most Great, Allah is Most Great, there is no god but Allah, Allah is Most Great, Allah is Most Great, and to Allah be praise).

2 – The second view is that it is:
Allahu akbar, Allahu akbar, Allahu Akbar, laa ilaaha ill-Allah, Allahu Akbar, Allahu akbar, Allahu akbar, wa lillaahi’l-hamd
(Allah is Most Great, Allah is Most Great, Allah is Most Great, there is no god but Allah, Allah is Most Great, Allah is Most Great, Allah is Most Great, and to Allah be praise).

3 – The third view is that it is:
Allahu akbar, Allahu akbar, Allahu Akbar, laa ilaaha ill-Allah, Allahu akbar, Allahu akbar, wa lillaahi’l-hamd
(Allah is Most Great, Allah is Most Great, Allah is Most Great, there is no god but Allah, Allah is Most Great, Allah is Most Great, and to Allah be praise).

The matter is broad in scope because there is no text narrated from the Prophet (sallallahu alaihi wassalam) which specifies a particular form of the Takbeer.

The time of the Takbeer
The Takbeer is divided into two categories:

1 – Takbeer at any time.
This is Takbeer that is not limited to a specific time, and it is Sunnah to say it all the time, morning and evening, before prayer and after prayer, at all times.

2 – Takbeer at certain times.
This is Takbeer which is limited to the time immediately after prayer. It is Sunnah to recite Takbeer at any time during the first ten days of Dhu’l-Hijjah and all the days of Tashreeq, starting from the beginning of the month of Dhu’l-Hijjah
(i.e., from sunset on the last day of Dhu’l-Qa’dah) until the end of the days of al-Tashreeq, which is when the sun sets on the thirteenth day of Dhu’l-Hijjah).

With regard to certain times, it starts from Fajr on the day of ‘Arafah and lasts until the sun sets on the last day of the days of Tashreeq, in addition to the Takbeer that may be recited at any time.

So when a person says the tasleem at the end of the obligatory prayer, he should pray for forgiveness (by saying Astaghfir-Allah) three times, and say, “Allahumma anta al-salaam wa minka al-salaam, tabaarakta yaa dhaa’l-jalaali wa’l-ikraam (“O Allah, You are al-Salaam [the One Who is free from all defects and deficiencies], and from You is all peace, blessed are You, O possessor of majesty and honor)”,
then start the Takbeer.

This is for those who are not on Hajj. The pilgrims should start the Takbeer recited after prayer from Zuhr on the Day of Sacrifice (the 10th of Dhu’l-Hijjah). And Allah knows best.

See Majmoo’ Fataawa Ibn Baaz (may Allah have mercy on him),
13/17; al-Sharh al-Mumti’
by Ibn ‘Uthaymeen, 5/220-224.