It is recommended that one recites Takbeer aloud in such days. As for the Muslim woman, she should not recite it a loud.
The Takbeer is divided into two categories:
1 – Takbeer at any time. This is Takbeer that is not restricted to be done after each of the five obligatory prayers. So, one can do it at all times and all places during the ten days of Thul-Hijjah and days of At-Tashreeq. Amongst the evidences on it are:
It was narrated on the authority of Ibn ‘Umar, may Allaah be pleased with them, that the Prophet, peace and blessing be upon him, said: “There are no days on which good deeds are greater or more beloved to Allaah than on these ten days, so recite much Tahleel (i.e. saying Laa ilaaha ill-Allaah (there is no god worthy of worship except Allaah)), Takbeer (i.e. saying Allaahu akbar) and Tahmeed (i.e. saying al-hamdu Lillaah (all perfect praise be to Allaah))”. [Reported by Imam Ahmad (2/131 and 75) and Abu ‘Awaanah and it is Hasan (good) Hadeeth because there are many routes and witnessing narrations to support it]
Moreover, it was narrated that ‘Abdullaah ibn ‘Amr, may Allaah be pleased with them, said: “The Messenger of Allaah, peace and blessing be upon him, used to go to The ‘Eid prayer while saying Tahleel and Takbeer loudly.” [Saheeh (authentic) Hadeeth based on the corroborating narrations. Refer to Al-Irwaa’ 3/123]
It was also narrated that Naafi‘ said: “Ibn ‘Umar, may Allaah be pleased with them, used to say Takbeer loudly until he reached the place of prayer when he goes to the prayers of ‘Eids Al-Fitr and Al-Adh-ha, then he would continue making Takbeer until the Imaam (i.e. the person who lead people in prayer) came out and he would join him in making Takbeer.” [Reported by Ad-Daaraqutni with an authentic Isnaad (i.e. chain of transmission)]
Moreover, Naafi‘ narrated that Ibn ‘Umar, may Allaah be pleased with them, used to recite Takbeer in Mina on these days, after every prayer, while reclining on his couch, in his tent, in his gatherings and while he was walking all through these days. [Reported by Ibn Al-Munthir in Al-Awsat with a good chain of transmission and Al-Bukhaari as a Mu‘allaq (suspended) Hadeeth]
2- Takbeer at certain times. This is Takbeer that is limited to the time immediately after prayer. It starts from Fajr on the day of ‘Arafah and lasts until the sun sets on the last day of the days of Tashreeq which is the thirteenth of Thul-Hijjah. It was narrated that Shaqeeq ibn Salamah , may Allaah have mercy upon him, said: “‘Ali, may Allaah be pleased with him, used to recite Takbeer after Fajr on the day of ‘Arafah and would not stop until the Imaam arrived on the last day of slaughtering at ‘Asr prayer.” [Reported by Ibn Al-Munthir, Al-Bayhaqi, and An-Nawawi and Ibn Hajar deemed it Saheeh (authentic) Hadeeth] The same was also reported of Ibn ‘Abbaas, may Allaah be pleased with them.
Ibn Taymiyyah, may Allaah have mercy upon him, said, “The most authentic ruling regarding Takbeer, which is the ruling of the majority of the scholars as well as the Companions is that it starts on the Day of ‘Arafah Fajr time and it lasts until the last day of slaughtering at ‘Asr time, being recited after each of the obligatory prayers.” [Majmoo‘ Al-Fataawa 24/20]
Ibn Hajar, may Allaah have mercy upon him, said: “The most authentic ruling regarding Takbeer, which is reported from the companions is the opinion of ‘Ali and Ibn Mas‘ood, may Allaah be pleased with them all, is that it starts on the Day of ‘Arafah Fajr time and it lasts until the end of the days of Mina. This was reported by Ibn Al-Munthir and others and Allaah Knows best.” [2/536]
The most authentic narration reported regarding the time of the specific Takbeer is the opinion reported from ‘Ali and Ibn ‘Abbaas, may Allaah be pleased with them, that it starts on the Day of ‘Arafah morning time and it lasts until the ‘Asr of the last day of slaughtering.” [Reported by Ibn Al-Munthir in Al-Awsat (2209-2210) and reported by Ibn Abi Shyabah in Al-Musnaf (1/488 - 489)].
This applies to those who are not on Hajj; for those who are on Hajj, they should start specific Takbeer after Zuhr prayer of the Day of sacrifice.
The most authentic wording of Takbeer is what was reported by ‘Abdur-Razzaaq with an authentic chain of transmission from Salmaan, may Allaah be pleased with him, who said: “Glorify Allaah saying: Allaahu Akbar, Allaahu Akbar, Allaahu Akbar Kabeera (Allaah is Most Great, Allaah is Most Great, Allaah is Most Great).” [Ibn Hajar, may Allaah have mercy upon him, deemed it authentic in Al-Fat-h (2/462)]
It was also authentically reported from ‘Umar and Ibn Mas‘ood, may Allaah be pleased with them, the following wording: “Allaahu Akbar, Allaahu Akbar, Laa ilaaha ill-Allaah, wal Allaahu akbar, Allaahu akbar, wa lillaahi’l-hamd (Allaah is Most Great, Allaah is Most Great, there is no god but Allaah, Allaah is Most Great, Allaah is Most Great, and to Allaah be praise).” [Reported by Ibn Al-Munthir in Al-Awsat (2207) from ‘Umar, may Allaah be pleased with him, and reported by Ibn Abi Shaybah (1/488 - 490) and Ibn Al-Munthir in Al-Awsat (2208) and At-Tabaraani in Al-Kabeer (9/ 307) and Az-Zu‘ayli deemed its chain of transmission Hasan in Nasb Ar-Raayah (2/224) from Ibn Mas‘ood, may Allaah be pleased with him].