Allah says in Suratul Hajj:

That they might witness things that are of benefit to them, and mention the name of Allah on appointed days, over the beast of cattle that He has provided for them (for sacrifice)...
[22:28]

The Messenger of Allaah (sallallaahu ‘alaihi wa sallam) said:

“No good deeds done on other days are superior to those done on these days [meaning the ten days of Dhul-Hijjah].” The companions asked, “O Messenger of Allaah, not even jihad in the way of Allaah?” He said, “Not even jihad, except for the man who puts his life and wealth in danger [for Allaah’s sake] and returns with neither of them.”
[Al-Bukhari]

He (sallallaahu ‘alaihi wa sallam) also said:

There is no day more honorable in Allaah’s sight and no acts more beloved therein to Allaah than those in these ten days. So say tahlil (There is no deity worthy of worship but Allaah : Laa ilaaha illallaah), takbir (Allaah is the greatest : Allaahu Akbar) and tahmid (All praise is due to Allaah : alhumdulillaah) a lot [on those days].
[Ahmad]

And:

There are no days more loved to Allaah for you to worship Him therein than the ten days of Dhul Hijja. Fasting any day during it is equivalent to fasting one year and to offer salatul tahajjud (late-night prayer) during one of its nights is like performing the late night prayer on the night of power [Lailatul Qadr].
[At-Tirmidhi]

On the Day of Nahr (10th of Dhul-Hijja), the Messenger (sallallaahu ‘alaihi wa sallam) stood in between the Jamrat during his Hajj and said:

This is the greatest Day (the 10th of Dhul-Hijjah) – the Day of Nahr (sacrifice).
[Al-Bukhari]

The Messenger of Allaah (sallallaahu ‘alaihi wa sallam) said:

Fasting on the day of ‘Arafah (9th) is an expiation for 2 years, the year preceding it and the year following it.
[Muslim]

The Prophet (sallallaahu ‘alaihi wa sallam) said:

The day of ‘Arafah, the day of sacrifice, and the days of tashreeq are ‘ids for us--the people of Islam--and they are days of eating and drinking [days of ‘id].
[At-Tirmidhi]

Abu Hurayrah radhi’allaahu ‘anhu said, “The Messenger of Allaah (sallallaahu ‘alaihi wa sallam) forbade fasting on the day of ‘Arafah for one who is actually at ‘Arafah.” And At-Tirmidhi comments: “The scholars prefer that the day of ‘Arafah be fasted unless one is actually at ‘Arafah.”

The Prophet (peace and blessings be upon him) said:

When you see the new moon of Dhul-Hijjah, if any one of you wants to offer a sacrifice, then he should stop cutting his hair and nails until he has offered his sacrifice.
[Muslim]

The Prophet sallallahu ‘alaihi wa sallam said:

There is no day on which Allaah frees people from the Fire more so than on the day of ‘Arafah. He comes close to those (people standing on ‘Arafah), and then He revels before His Angels saying: “What are these people seeking?”
[Muslim]

Action List:

1. Know that the days of Dhul-Hijjah, the first 10 days, the Day of ‘Arafah (9th), the day of Nahr (10th) up to the 13th (Tashreeq) are sacred days.

2. In general do as many good deeds as you can in these blessed days including asking for forgiveness.

3. Especially say “laa ilaaha illallaah”, “alhamdulillah” and “allahu akbar”.

4. Fasting 1 day in these 10 days is like fasting a year and 1 tahajjud prayer is like praying tahajjud on Laylatul Qadr.

5. Fast on the Day of ‘Arafah (9th of Dhul Hijjah) as it expiates 2 years of sin- the one before and after.

6. Make your sacrifice for Allah (10th - the day of Nahr) - do not cut your hair or nails from the 1st of Dhul-Hijjah if you intend to make the sacrifice.

7. Celebrate ‘Eid ul-Adhaa (11th, 12th, 13th) and join the prayer with your community (11th).

8. Especially make takbeer from Fajr on the Day of ‘Arafah (9th) until after ‘Asr on the last Day of Tashreeq (13th).  The takbeer should be recited loudly whenever possible to remind everyone and to glorify Allah’s praises.