The best days to fast in Muharram is the day of ‘Aashoora and one day before it or one day after it. Imam Shafi’i and imam Ahmad both stated, “It is mustahabb [recommended] to fast both the ninth and the tenth, because the Prophet ṣallallāhu ‘alayhi wa sallam fasted the tenth and intended to fast the ninth.”
Shaykh Al Munajjid gives a tip for us to be sure that we fasted on the correct day:
If a Muslim wants to be sure that he has fasted on the right day, he should fast two consecutive days at ‘Aashooraa’. So he should calculate when ‘Aa’shooraa’ will be if Dhu’l-Hijjah is twenty-nine days and if it is thirty days, and fast these two days. Thus he will be definite that he has fasted ‘Aashooraa’, and in this case he will have fasted either the ninth and tenth, or the tenth and eleventh, both of which are good. If he wants to be sure of fasting Taasoo’ah (the ninth of Muharram) as well, then he should fast the two days we have spoken of above and the day immediately before them as well. Then he will have fasted the ninth, tenth and eleventh, or the eighth, ninth and tenth. In either case he will have fasted the ninth and tenth for sure.
Fasting the “White Days.” The Prophet ṣallallāhu ‘alayhi wa sallam said, “Fasting three days of each month is fasting for a lifetime, and ‘the white days’ are the thirteenth, fourteenth and fifteenth.” [an-Nasa’i, Saheeh] Ibn ‘Abbas raḍyAllāhu ‘anhu said, “The Messenger of Allah did not fail to fast the white days either when at home or on a journey.” [an-Nasa’i] These days are the 13th, 14th and 15th of the hijri month which you can calculate according to whichever calendar for Muharram you follow insha’Allah.
Increasing in Du’a while fasting
“The prayer (du’a) of the fasting person will not be refused.” [Al-Bayhaqi, Saheeh]
The Prophet ṣallallāhu ‘alayhi wa sallam said in the hadith, “and the best prayer after the obligatory prayer is prayer at night.” [Muslim]
Protecting Yourself from Sins
Allah azza wa jal said about the sacred months, “so wrong not yourselves therein” (9:36) imam Sa’di rahimahullah said in his tafseer regarding this ayah, Allah states that He has made them a measure of time for His slaves, which they may use for worshipping Him, and thank Allah for His blessings, and they serve the interests of His slaves, so beware of wronging yourselves therein. The pronoun may also be understood as referring to the four sacred months, and this forbids them to wrong themselves in those months in particular, as well as it being forbidden to do wrong at all times, because it is more forbidden at this time, but it is worse at this time than at others.
Increasing in Good Deeds in general
Read some chapters from Riyaad as-Saaliheen for ideas.
May Allah subḥānahu wa ta’āla accept our deeds and allow us to reap the benefits of those days and times He has selected over others.