Dear brother:
 Soon we will have the blessed month Dhi al Hijjah [the eleven month of the lunar calendar where Muslims perform Hajj]. The first ten days of Dhi- al-Hijja are the best days of this life. The Prophet (PBUH) said about them as narrated by Ibn Abass:

“There are no days in which the righteous deed is more pleasing to Allah the Almighty than these days, meaning the ten days, they said: “Messenger of Allah, not even the jihad for the sake of Allah?” He said: “not even the jihad for the sake of Allah, except for a man who went out with nothing but himself and his money and never came back with anything.”

[Al-Bukhari reported]

 Ibn Umar narrated that the Prophet (PBUH) said,

“There are no days in which deed is greater and more beloved to Allah the Almighty than the ten days, so do more (tahlil) exaltation, (Tahmid) praising, and glorification (Takbir)”

[Reported by Ahmad]

 Allah enjoins remembering Him alot in these days.

((… And mention the Name of Allah on appointed days, over the beast of cattle that He has provided for them (for sacrifice). Then eat thereof and feed therewith the poor who have a very hard time… ))

(Al-Hajj: 28)

 Most scholars consider these known days are the first tendays of Dhi Hijjah. They include the day of 'Arafah; which is the day of Greatest Hajj, in which sins are forgiven, and fasting it:

((…expiates [the sins] of the past year and the present one…))

[Reported by Muslim]

 In addition, the include the day of an-Nahr (slaughtering). It is the tenth day of Dhi Hijjah. It is one of the best days to Allah. The Prophet (PBUH) said:

((The greatest day in Allah's sight (SWT) is the day of An-Nahr then the day of al-Qerr))

 The day of al-Qerr is the day after theday of An-Nahr. As for deeds preferable to do during these days, they are:
 1- Repentance and quitting all disobediences [actions] and sins.
 2- Performing the prayer: is one of the greatness and most rewarded deeds. Therefore, a Muslim must take care of it, on time and in congregation. He should also perform a lot of nawafil (voluntary prayer) during these days.
 3- Fasting, and specifically the day of 'Arafa. Abu Qtada narrated that the Prophet (PBUH) said,

((Fasting the day of 'Arafah, will be credited by Allah with, forgiving the [minor] sins of previous and the coming year)).

[Muslim]

 Hafsah [one of the prophet wives] said:

((There are four things that the Messenger (PBUH) never abandoned; fasting the day of 'Ashurah, fasting the [first] ten [days of Dhi-Hijjah], fasting three days of every month and praying two Rak'ahs before the dawn prayer.))

[On the authority of Ahmad and An-Nasa'i]

 4- Doing alot of; takbeer (Saying that Allah is the greater), tahmeed (praising, tahlil (saying; there is no God but Allah), and Remembrance (remembrance of Allah). Al-Bukhari reported that, Ibn Umar and Abu Huraira (MABPWT) used to go to the market during the ten days, saying Takbeer, and then people said the same after them.
 5- Doing alot of righteous deeds.