Takbir on Dhul-Hijjah Days: What to Say and When?

Question: As-salamu `alaykum. When should Muslims start takbir on Dhul-Hijjah, and what is the exact wording transmitted from the Prophet (peace and blessings be upon him) in this regard?

Wa `alaykum as-Salamu wa Rahmatullahi wa Barakatuh.
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
First of all, it is important to note that takbir is permissible from the beginning of Dhul-Hijjah until the end of the 13th day.
There are two types of takbir: unrestricted and restricted. Unrestricted takbir starts from the beginning of Dhul-Hijjah until the days of `Eid, while restricted takbir is confined to the time after the obligatory Prayers.
The transmitted wording of takbir is,
Allahu Akbar, Allahu Akbar, la Ilaha illa Allah; Allahu Akbar, Allahu Akbar, walillah al-hamd.
(Allah is the Greatest, Allah is the Greatest. There is no god but Allah; Allah is the Greatest, Allah is the Greatest, and all praise is due to Allah).
There is also another version transmitted from Salman (may Allah be pleased with him), who used to say, “Allahu Akbar, Allahu Akbar, Allahu Akbar Kabira” (Allah is the Greatest, Allah is the Greatest, Allah is the Ever Greatest). Then he followed this by sending blessings on the Prophet (peace and blessings be upon him). This version of takbir is applicable at any time; however, it was not transmitted from the Prophet (peace and blessings be upon him) nor from any other righteous companions (may Allah be pleased with them).
There are two types of takbir in Dhul-Hijjah: unrestricted takbir and restricted takbir. Unrestricted takbir is applicable at any time or place, from the first of Dhul-Hijjah until the days of `Eid. Hence, a man can make takbir in the roads, markets, in Mina, and also when people meet each other. All these are occasions for unrestricted takbir.
Restricted takbir, on the other hand, is restricted to the time following every obligatory prayer, especially if it is offered in congregation, as most scholars restrict it. This type of takbir also includes takbir in the place of `Eid Prayer, on the way to it and while one is sitting around waiting for the `Eid prayer.
On such occasions, one should not remain silent, in both `Eid Al-Fitr or `Eid Al-Adha, for these are days on which Islamic rites should be properly observed.
Takbir is one of the most prominent Islamic rites. It is reported that the Prophet (peace and blessings be upon him) said, “Ornament your feasts with takbir.” (At-Tabarani)
Therefore, Muslims should observe takbir on the day of Eid. On their way to perform Eid Prayer and as they wait for it, Muslims should make takbir loudly.
Allah Almighty knows best.