The excellence of these ten days have been mentioned in the Qur'an and the Sunnah.
1. Allah says in the Qur'an:
"By the dawn and by the ten nights … " [Al-Qur'an 89:1-2]
Ibn Kathir said that "the ten nights" referred to here are the ten days of Dhul-Hijjah, and this opinion was also held by Ibn Abbas, Ibn az-Zubair, Mujahid and others.
2. The Messenger of Allah, peace be upon him, said:
"There are no deeds as excellent as those done in these ten days." They (the companions listening) said, "Not even Jihad?" He, peace be upon him, said, "No, not even Jihad except a man who goes forth endangering his life and wealth and does not return with anything." [Recorded by Imam al-Bukhari]
3. Allah says:
" … and remember the name of Allah in the appointed days." [Al-Qur'an 2:203]
Ibn Abbas and Ibn Kathir said this means in the ten days of Dhul-Hijjah.
4. The Messenger of Allah, upon whom be peace, said:
"There are no other days that are as great as these in the sight of Allah, the Most Sublime. Nor are there any deeds more beloved to Allah then those that are done in these ten days. So increase in tahlil (to say la illaha illallah), takbir (to say allahu akbar) and tahmid (to say alhumdulillah)." [Reported by at-Tabarani in al-Mu'jum al-Kabir]
Rasul-ullah (Sallaho Alaihe Wassallam) used to fast all 9 days including Arafah:
حدثنا مسدد حدثنا أبو عوانة عن الحر بن الصباح عن هنيدة بن خالد عن امرأته عن بعض أزواج النبي صلى الله عليه وسلم قالت كان رسول الله صلى الله عليه وسلم يصوم تسع ذي الحجة ويوم عاشوراء وثلاثة أيام من كل شهر
One of the wives of the Prophet (peace and blessings be upon him) said: "Allah's Messenger used to fast the [first] nine days of Dhul-Hijjah, the day of 'Ashurah, and three days of each month." (Reported by Abu Dawud).
Fasting on Arafah day is an expiation for sins committed the year before and the year after:
حدثنا أحمد بن عبدة أنبأنا حماد بن زيد حدثنا غيلان بن جرير عن عبد الله بن معبد الزماني عن أبي قتادة قال قال رسول الله صلى الله عليه وسلم صيام يوم عرفة إني أحتسب على الله أن يكفر السنة التي قبله والتي بعده
Abu Qatadah (may Allah be pleased with him) is reported to have said that the Prophet (peace and blessings be upon him) said: "Fasting on the day of 'Arafah is an expiation (of sins) for two years, the year preceding it and the year following it. (Reported by Ibn Majah)
Sacrificing an animal is an established Sunnah on Eid day:
حدثنا حجاج قال حدثنا شعبة قال أخبرني زبيد قال سمعت الشعبي عن البراء قال سمعت النبي صلى الله عليه وسلم يخطب فقال إن أول ما نبدأ من يومنا هذا أن نصلي ثم نرجع فننحر فمن فعل فقد أصاب سنتنا
Narrated Al-Bara' bin 'Azib (RA) that I heard the Prophet (Sallaho Alaihe Wassallam) say when he gave a sermon (on the day of Idal-Adha), "The first thing we will do on this day of ours, is to offer the ('Id) prayer and then return to slaughter the sacrifice. Whoever does so, he acted according to our Sunna (tradition). (Reported by Bukhari)
Eating, drinking, enjoying and reciting Ayamul-Tashreeq: 11th, 12th & 13th of Dhul-Hijjah:
وَاذْكُرُوا اللَّهَ فِي أَيَّامٍ مَعْدُودَاتٍ فَمَنْ تَعَجَّلَ فِي يَوْمَيْنِ فَلَا إِثْمَ عَلَيْهِ وَمَنْ تَأَخَّرَ فَلَا إِثْمَ عَلَيْهِ لِمَنِ اتَّقَى وَاتَّقُوا اللَّهَ وَاعْلَمُوا أَنَّكُمْ إِلَيْهِ تُحْشَرُونَ
[2:203] Recite the name of Allah during the given number of days. Then whoever is early in leaving after two days, there is no sin on him, and whoever leaves later, there is no sin on him, if he is God-fearing. Fear Allah and be sure that you are going to be gathered before Him.
قَالَ اِبْن عَبَّاس : الْأَيَّام الْمَعْدُودَات أَيَّام التَّشْرِيق وَالْأَيَّام الْمَعْلُومَات أَيَّام الْعَشْر . وَقَالَ عِكْرِمَة " وَاذْكُرُوا اللَّه فِي أَيَّام مَعْدُودَات " يَعْنِي التَّكْبِير فِي أَيَّام التَّشْرِيق بَعْد الصَّلَوَات الْمَكْتُوبَات : اللَّه أَكْبَر اللَّه أَكْبَر . وَقَالَ الْإِمَام أَحْمَد : حَدَّثَنَا وَكِيع حَدَّثَنَا مُوسَى بْن عَلِيّ عَنْ أَبِيهِ قَالَ : سَمِعْت عُقْبَة بْن عَامِر قَالَ : قَالَ رَسُول اللَّه - صَلَّى اللَّه عَلَيْهِ وَسَلَّمَ - " يَوْم عَرَفَة وَيَوْم النَّحْر وَأَيَّام التَّشْرِيق عِيدُنَا أَهْل الْإِسْلَام وَهِيَ أَيَّام أَكْلٍ وَشُرْبٍ
Ibn Abbas (RA) said, "The given number of days are the Ayamul-Tashreeq And the known days are the ten. Ikramah (RA) said, "Remember Allah (SWT) in the given number of days i.e. recite the Takbeer (Allahu Akbarm Allahu Akbar) in Ayamul-Tashreeq after the Fardh (obligatory) Prayers). Uqbah Ibn Amir (RA) narrates that Rasul-ullah (Sallaho Alaihe Wassallam) said that the day of Arafah, the day of Sacrifice and the Ayanul-Tashreeq are our days (for people of Islam) and eat and drink in these days. (Tafseer Ibn Katheer)
فَقَالَ عُمَر بْن الْخَطَّاب وَعَلِيّ بْن أَبِي طَالِب وَابْن عَبَّاس : ( يُكَبِّر مِنْ صَلَاة الصُّبْح يَوْم عَرَفَة إِلَى الْعَصْر مِنْ آخِر أَيَّام التَّشْرِيق
Umar Ibn Khattab (RA), Ali Ibn Abi Talib (RA) & Ibn Abbas (RA) said, "Recite the Takbeer from the Fajar of the day of Arafah until the Asar of the last day of Ayamul-Tashreeq (13th of Dhul-Hijjah). (Tafseer of Imam Qurtubi)
وَفِي الْمُخْتَصَر عَنْ مَالِك : اللَّه أَكْبَر اللَّه أَكْبَر , لَا إِلَه إِلَّا اللَّه , وَاَللَّه أَكْبَر , اللَّه أَكْبَر وَلِلَّهِ الْحَمْد
And in Summary it is narrated from Imam Malik (RA) [about the Takbeer] Allahu Akbar, Allahu Akbar, La Ilaha Illallah, Wallahu Akbar Allahu Akbar Wallilahil Hamd
Takbeer Tashreeq is the recitation of Allahu akbar, La ilaha illa hu Wallahu akbar, Allahu akbar wa lillahil hamd after every fardh salaah commencing from the Fajr of the 9th of Dhul Hijah until after Asr of the 13th of Dhul Hijjah. (Shami vol. 1 pg. 406)
When Ibrahim (AS) began moving the knife on his beloved son, Ismaeel (AS), the angels sent by Allah with a ram exclaimed. Allahu akbar, Allahu akbar (Allah is the greatest, Allah is the greatest). Ibrahim (AS) heard the voice of the angels and replied, La illah illa Allahu Wallahu akbar (There is no god besides Allah, and Allah is the greatest).
His son Ismaeel (AS) heard this conversation and understood that Allah had relieved him from this great rial, thus he replied, Allahu Akbar Walillahil hamd (Allah is the greatest, and to Allah belongs all praise)(Ibid).
Mustahab (Prefferable) NOT to cut any hair or trim any nails if one is preparing to make a sacrifice for Eid:
حدثنا ابن أبي عمر المكي حدثنا سفيان عن عبد الرحمن بن حميد بن عبد الرحمن بن عوف سمع سعيد بن المسيب يحدث عن أم سلمة أن النبي صلى الله عليه وسلم قال إذا دخلت العشر وأراد أحدكم أن يضحي فلا يمس من شعره وبشره شيئا قيل لسفيان فإن بعضهم لا يرفعه قال لكني أرفعه
Umm Salamah (May Allah be pleased with her) said: The Messenger of Allah (PBUH) said, "When anyone of you intends to sacrifice the animal and enter in the month of Dhul-Hijjah, he should not get his hair cut or nails pared till he has offered his sacrifice.'' (Reported by Muslim)
Note: Ulama have stated that this includes moustache and hair of the private parts etc. However, if the hair under the arms and in the pubic region is very long (which renders the Salaat Makrooh i.e. more than forty days), then it will become compulsory to remove such hair. However, if a person forgets to clip the nails before the 1st of Dhul Hijjah and the nails have grown so long that they may cause injury, then he may clip them.
2. Fast all nine days and especially on the ‘Day of Arafah’
The Prophet peace be up on him, pbuh said: “Anyone who fasts for one day for Allah’s pleasure, Allah will keep his face away from the (Hell) fire for (a distance covered by a journey of) seventy years.” (Bukhari, Muslim)
The Prophet peace be up on him, pbuh used to fast on the ninth day of Dhul-Hijja and he said: “Fasting the Day of ‘Arafah (ninth Dhul-Hijja) is an expiation for (all the sins of) the previous year and expiation for (all the sins of) the coming year.” (Muslim)
3. Perform Dhikr and Takbeer
The Prophet peace be up on him, pbuh said: “There are no days on which good deeds are greater or more beloved to Allah than on these ten days, so recite much Tahleel (saying Laa ilaaha ill-Allah), Takbeer (saying Allahu Akbar) and Tahmeed (saying Alhamdu Lillaah).” (Ahmad -Saheeh)
Ibn ‘Umar and Abu Hurayrah used to go out in the marketplace during the first ten days of Dhul-Hijja, reciting Takbeer, and the people would recite Takbeer when they heard them. (Bukhari)
Takbeer at this time is a Sunnah that has mostly been forgotten, especially during the first few days. So say it in the masjid, in your home, on the street and every place where it is permitted to remember Allah. Revive the Sunnah that have been virtually forgotten and earn great rewards for doing so.
The Prophet peace be up on him, pbuh said: “Whoever revives an aspect of my Sunnah that is forgotten after my death, he will have a reward equivalent to that of the people who follow him, without it detracting in the least from their reward.” (Tirmidhi- Hasan)