At the end of Ramadaan Allaah has prescribed that His slaves should recite takbeer. Allaah says (interpretation of the meaning): 

“(He wants that you) must complete the same number (of days), and that you must magnify Allaah [i.e. to say Takbeer (Allaahu Akbar: Allaah is the Most Great)] for having guided you so that you may be grateful to Him”

[al-Baqarah 2:185]

“that you must magnify Allaah” means venerating Him in your hearts and on your lips, by pronouncing the takbeer. 

So you should say: Allaahu akbar, Allaahu akbar, laa ilaaha ill-Allaah, Allaahu akbar, Allaahu akbar, wa Lillaahi’l-hamd (Allaah is Most Great, Allaah is Most Great, there is no god except Allaah, Allaah is Most Great, Allaah is Most Great, and all praise be to Allaah). 

Or you can say Allaahu akbar three times, so you say: 

Allaahu akbar, Allaahu akbar, Allaahu akbar, laa ilaaha ill-Allaah, Allaahu akbar, Allaahu akbar, Allaahu akbar, wa Lillaahi’l-hamd (Allaah is Most Great, Allaah is Most Great, Allaah is Most Great, there is no god except Allaah, Allaah is Most Great, Allaah is Most Great , Allaah is Most Great, and all praise be to Allaah). 

Both are permissible. 

This takbeer is Sunnah according to the majority of scholars. It is Sunnah for both men and woman, in the mosque, in the home and in the marketplace. Men should say it out loud, and women should say it quietly, because women are commanded to lower their voices. Hence the Prophet (peace and blessings of Allaah be upon him) said: “If you notice something during the prayer, men should say ‘Subhan-Allaah!’ and women should clap.” So women should say the takbeer quietly and men should say it out loud. 

The time for this takbeer begins when the sun sets on the night of Eid (i.e., the night before) if it is known that the new month has begun when the sun sets, such as when the people have completed thirty days of Ramadaan, or when it is proven that the new moon of Shawwaal has been sighted. And it ends when the prayer begins, i.e., when they start the Eid prayer then the time for the takbeer ends. 

Majmoo’ Fataawa Ibn ‘Uthaymeen, 16/269-272. 

Al-Shaafa’i said in al-Umm: 

Allaah says concerning the month of Ramadaan (interpretation of the meaning):

“(He wants that you) must complete the same number (of days), and that you must magnify Allaah [i.e. to say Takbeer (Allaahu Akbar: Allaah is the Most Great)] for having guided you”

[al-Baqarah 2:185]

I heard from some scholars whom I trust and who are knowledgeable about the Qur’aan: 

“(He wants that you) must complete the same number (of days), and that you must magnify Allaah [i.e. to say Takbeer (Allaahu Akbar: Allaah is the Most Great)] for having guided you” it is completed when the sun sets on the last day of the month of Ramadaan. 

Then al-Shaafa’i said: 

When they see the new moon of Shawwaal, I like all the people to recite takbeer together and individually in the mosques, the marketplaces, the streets, the houses, travellers and non-travellers in all situations, no matter where they are. They should pronounce the takbeer openly and continue to recite it until they come to the prayer-place the next morning, and after that until the imam comes out to lead the prayer, then they should stop reciting the takbeer. 

Then he narrated from Sa’eed ibn al-Musayyib, ‘Urwah ibn al-Zubayr, Abu Salamah and Abu Bakr ibn ‘Abd al-Rahmaan, that they used to recite takbeer on the night before Eid al-Fitr in the mosque, reciting the takbeer out loud. 

It was narrated from ‘Urwah ibn al-Zubayr and Abu Salamah ibn ‘Abd al-Rahmaan that they used to recite the takbeer out loud when they came to the prayer-pace in the morning, and that Naafi’ ibn Jubayr used to recite the takbeer out loud when he came to the prayer-place in the morning of the day of Eid. 

It was narrated from Ibn ‘Umar that he used to come to the prayer-place on the morning of the day of Eid al-Fitr, when the sun had risen, and he would recite takbeer until he reached the prayer-place, then he would recite takbeer in the prayer-place until the imam sat down and stopped reciting takbeer.