Bismillahi Rrahmanni Rraheem.
Praise be to Allaah.
Ten days are better than all other days of the year, with no exception, even the last ten days of Ramadaan. But the last ten nights of Ramadaan are better than the first ten nights of Dhu'l-Hijjah, because they include Laylat al-Qadr, which is better than a thousand months. See Tafseer Ibn Katheer, 5/412.
So the Muslim should start these days with sincere repentance to Allaah, then he should do a lot of righteous deeds in general, then he should pay attention to doing the following deeds:
1 -- Fasting
It is Sunnah for the Muslim to fast on the first nine days of Dhu'l-Hijjah, because the Prophet (peace and blessings of Allaah be upon him) encouraged us to do righteous deeds in the first ten days of Dhu'l-Hijjah, and fasting is one of the best of deeds, which Allaah has chosen for Himself as it says in the hadeeth qudsi: "Allaah says: 'All the deeds of the son of Adam are for him, except fasting, which is for Me and I shall reward for it." Narrated by al-Bukhaari, 1805.
The Prophet (peace and blessings of Allaah be upon him) used to fast on the first nine days of Dhu'l-Hijjah. It was narrated from Hunaydah ibn Khaalid from his wife, that one of the wives of the Prophet (peace and blessings of Allaah be upon him) said: The Prophet (peace and blessings of Allaah be upon him) used to fast on the first nine days of Dhu'l-Hijjah and the day of 'Ashoora', and three days each month, the first Monday of the month and two Thursdays. Narrated by al-Nasaa'i, 4/205 and by Abu Dawood; classed as saheeh by al-Albaani in Saheeh Abi Dawood, 2/462.
2 -- Reciting tahmeed, tahleel and takbeer a great deal
"Allaahu Akbar, Allaahu Akbar, Allaahu Akbar, Laa ilaaha illa-Allaah,
Allaahu Akbar, 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 and to Allaah be praise).
It is Sunnah to recite takbeer (saying Allaahu akbar (Allaah is most great)), tahmeed (saying al-hamdu Lillah (praise be to Allaah)), tahleel (saying Laa ilaaha ill-Allaah (There is no god but Allaah)) and tasbeeh (saying Subhaan-Allaah (Glory be to Allaah)), during the first ten days of Dhu'l-Hijjah, saying these words out loud in the mosques, homes, streets and every place in which it is permissible to remember Allaah, to perform this act of worship openly and proclaim the greatness of Allaah.
Men should recite out loud and women should recite quietly.
Allaah 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 Allaah on appointed days, over the beast of cattle that He has provided for them (for sacrifice)" [al-Hajj 22:28]
The majority are of the view that the "appointed days" are the first ten days of Dhu'l-Hijjah, because it was narrated from Ibn 'Abbaas (may Allaah be pleased with him) that the appointed days are the first ten days of Dhu'l-Hijjah.
It was narrated from 'Abd-Allaah ibn 'Umar (may Allaah be pleased with him) that the Prophet (peace and blessings of Allaah be upon him) said: "There are no days that are greater before Allaah or in which good deeds are more beloved to Him, than these ten days, so recite a great deal of tahleel, takbeer and tahmeed during them." Narrated by Ahmad, 7/224; its isnaad was classed as saheeh by Ahmad Shaakir.
And there are other versions.
This takbeer has become a forsaken Sunnah nowadays, especially at the beginning of these ten days. You can hardly hear it except from a few. It should be recited out loud, so as to revive the Sunnah and remind the heedless. It was proven that Ibn 'Umar and Abu Hurayrah (may Allaah be pleased with them both) used to go out to the marketplace during the first ten days of Dhu'l-Hijjah, reciting the takbeer out loud, and the people would recite takbeer when they heard them, i.e., the people would remember the talkbeer and each one would recite it by himself; it does not mean that they recited in unison -- that is not prescribed in sharee'ah.
3 -- Doing Hajj and 'Umrah
One of the greatest deeds that can be done during these ten days is Hajj to the sacred House of Allaah. Whoever Allaah helps to go on Hajj to His House and do the rituals in the required manner will -- in sha Allaah -- be included in the words of the Messenger of Allaah (peace and blessings of Allaah be upon him): "An accepted Hajj brings no less a reward than Paradise."
4 -- Udhiyah (sacrifice)
One of the righteous deeds that bring a person closer to Allaah in these ten days is to slaughter the sacrifice and to look for a good, fat animal and to spend money on it for the sake of Allaah.
Let us hasten to make the most of these great days, before the negligent regrets what he did, and before he asks to go back and receives no response.