Generally speaking any act that Allah is pleased with should be done in these days. The primary act to be done abundantly is the tahleel, takbeer and tahmeed as these were mentioned in the narration of Ibn ‘Umar raḍyAllāhu 'anhu (may Allāh be pleased with him) (quoted below)

Dhikr [Remembrance of Allah swt]
Ibn Umar narrates that the Prophet (Peace be upon Him) said: “There are no days during which good deeds are greater are beloved to Allah than these ten days (Dhul Hijjah). So recite more often the tahleel, takbeer and tahmeed.”[Reported by Imam Ahmed].

Tahleel is to declare the oneness of Allah by saying La ilaaha il-lal-laah (There is none worthy of worship except Allah)

Takbeer is to proclaim the greatness of Allah by saying Allahu Akbar (Allah is Great!)

Tahmeed is to praise Allah by saying Alhamdulillah (All Praise belongs to Allah)

Tasbeeh is to glorify Allah by saying Subhanallah (Glory be to Allah)

Fasting
Abu Qataadah raḍyAllāhu 'anhu (may Allāh be pleased with him) reported that Rasullullah (Peace be upon Him) said: “I have hope in Allah Ta’ala that the fast of Arafaat will be an atonement for the sins of the past and the forthcoming year.” (Tirmidhi)
Abu Qatadah raḍyAllāhu 'anhu (may Allāh be pleased with him) narrates that the Prophet (Peace be upon Him) said regarding the fast of the day of Arafah, “It expiates the sins of the past year and coming year.” [Sahih Muslim]

It is reported from Hunaydah ibn Khalid that one of the wives of the Prophet (Peace be upon Him) stated that the Prophet (Peace be upon Him) used to fast the first nine days of Dhul Hijjah [Al-Nasai and Abu Dawood].

Sacrifice
Sacrifice, for it is the ancient ritual of our forefather Ibrahim and was ordered for our Prophet (Peace be upon Him) when Allah said "So pray to your Lord and sacrifice [to Him alone]” [108: 2]
The Prophet (Peace be upon Him) said: “The son of Adam does not perform any actions on the day of sacrifice which is more pleasing to Allah than the shedding of blood. The sacrificed animal shall come on the Day of judgment with its horns, hair, and hooves. The sacrifice is accepted by Allah before the blood reaches the ground…” [Tirmithi, Ibn Majah]
Regarding the sacrifice, Allah says in the Qur’an: “It is not their meat, nor their blood, that reaches Allah, It is their piety that reaches Allah.” [Surah Al-Haj:37]

Takbeers:
The Takbeer at-Tashreeq at least once after every fardh prayer from Fajr of the 9th of Zulhijjah until Asr time of the 13th of Zulhijjah.
It is: "Allahu akbar, Allahu akbar, la ilaha ill-Allah; wa Allahu akbar. Allahu akbar, wa Lillahi’l-hamd (Allah is Most Great, Allah is Most Great, there is no god but Allah; Allah is Most Great and to Allah be praise),”
Men are encouraged to recite these phrases out loud and women quietly

Doing more good deeds than usual

1. Prayer
Increase greatly in your prayer (Salah) for it is related, the Prophet (Peace be upon Him) said: “There are no days in which righteous deeds are more beloved to Allah than these ten days (Dhul Hijjah).” [Sunan Ibn Majah]

2. Recitation
Recite the Quran for it is the best form of Dhikr and the shortest route to draw closer to Allah. Set yourself an aspirational target for these days as you did set yourself a target for Ramadan.

An authentic hadith in At-Tirmithee states: Whoever reads a letter from the Book of Allah, he will have a reward. And that reward will be multiplied by ten. I am not saying that “Alif, Laam, Meem” is a letter, rather I am saying that “Alif” is a letter, “laam” is a letter and “meem” is a letter.” So increase your recitation of the Qur’an to gain these merits, and to gain the following merit as well.

In another hadeeth, ‘Aisha, raḍyAllāhu 'anhu (may Allāh be pleased with her) relates that the Prophet (Peace be upon Him) said: Verily the one who recites the Qur’an beautifully, smoothly, and precisely, he will be in the company of the noble and obedient angels. And as for the one who recites with difficulty, stammering or stumbling through its verses, then he will have TWICE that reward.” [Al-Bukhari and Muslim]

3. Charity
Donate in charity for charity has been linked to bravery while miserliness has been linked to cowardice. The Prophet (Peace be upon Him) would invoke, “O Allah, I seek refuge in you from miserliness and cowardice.”

4. Sadaqa
Help the needy (start from family and friends around you, those that will ask and those that will not due their level of bashfulness)

Abu Huraira raḍyAllāhu 'anhu (may Allāh be pleased with him) narrated that the Prophet (Peace be upon Him) said, ” … whoever alleviates the need of a needy person, Allah will alleviate his needs in this world and the Hereafter… [Sahih Muslim (4/2074) No. 2699, Sunan Tirmidhi (5/195) No. 2945, Sunan Ibn Majah (1/152) No. 225, Musnad Ahmad (2/252) No. 7421]

Conclusion
In summary, it is the responsibility of every Muslim, man and woman, old and young, to revive these ten days until they become for our community like the last ten nights of Ramadan.
Our lives are short and our accountability great. We cannot afford to let golden moments like these pass by without expending ourselves in the worship of Allah.

Allah Knows Best

May Allah Suhanahu wa ta’ala grant us goodness. Aameen.