The Holy Prophet sallallahu alayhi wasallam has exhorted and encouraged his ummah to fast on this day. Abu Qataadah radhiyallahu anhu narrates that the Holy Prophet sallahllahu alayhi wasallam was asked regarding the fast of the day of Aashoorah. The Holy Prophet sallallahu alayhi wasallam replied: "It is a compensation for the (minor) sins of the past year". (Muslim Vol.1 Page 368)

NOTE:- It must be borne in mind that wherever a promise of forgiveness of sins is made upon the fulfillment of some action, it is only the minor sins that are forgiven. The major sins are not forgiven without true tawbah (repentance).

One should also observe the fast of the 9th Muharram to safeguard his deed from resemblance with the non-muslims who fast only on the 10th Muharram. Ibne Abbas radhiyallahu anhu said: When Rasulullah sallallahu alayhi wasallam observed the fast of the day of Aashoorah and ordered (his Companions) to fast, they said, "O Rasulullah!It is a day revered by the Jews and Christians." Rasulullah sallallahu alayhi wasallam said, "The coming year, if Allah wills, we will fast on the ninth (also)". (Muslim Vol.1
Page 359)

These ahadeeth indicate clearly that one should fast on the ninth and tenth of Muhaaram. However, if one does not manage to fast on the ninth, then he should fast on the eleventh of Muharram instead. The Holy Prophet sallallahu alayhi wasallam said: "Observe the fast of Aashoorah and oppose the Jews. Fast a day before it or a day after." (Bayhaqi). Hence, it is important to either fast on the ninth and tenth of Muharram or the tenth and eleventh. To fast only on the day of Aashoorah is makrooh tanzeehi as stated by Allamah Ibne Aabideen Shaami.
One should be generous on one’s family and dependents and spend more on them than what is normally spent. Abu Hurayrah radhiyallahu anhu reports that
Rasulullah sallallahu alayhi wasallam said: "One who
generously spends on his family on the day of Aashoorah,
Allah will increase (his provision) for the whole year". (Bayhaqi, At Targheeb Wat Tarheeb Vol. 2 Page 116)

These are the only actions supported by ahaadeeth.