Answer to those who ask "Am I responsible for my previous actions after I start to live by Islam?"

According to the Qur'an, people who do not know about Islam are "ignorant" and cannot distinguish between what is right and what is wrong. Once they embrace Islam, however, they obtain this knowledge, leave their former un-Islamic ways behind, ask for Allah's forgiveness, and repent. As a manifestation of Allah's infinite mercy, He promises to accept all people's repentance and transform their wrong actions into good, provided that they sincerely turn to Him, repent for their sins, and follow the righteous path:

… [E]xcept for those who repent and believe and act rightly: Allah will transform the wrong actions of such people into good—Allah is Ever-Forgiving, Most Merciful (Surat al-Furqan, 70)

But I am Ever-Forgiving to anyone who repents and has faith and acts rightly and then is guided. (Surat al-Anbiya', 82)

Repentance is an act of worship that is not unique to new Muslims. In reality, it is a lifetime worship performed by all Muslims, one that guides them to Allah's infinite mercy and forgiveness, for no believer is immune to committing sins and making mistakes.
In Allah's presence, believers strive to maintain a balance between hope and fear. Given that they cannot know their situation until they die and face Allah on the Day of Judgment, there is no way for them to feel secure against Allah's punishment. This aside, no matter how desperate their situation or how great their sin, they do not despair of Allah's mercy.
One of satan's greatest tricks is to infect people with hopelessness by causing them to believe such things as: "It is too late," "No one can save you now," or "You have already immersed yourself in sin." By insinuating such false claims, satan and his followers seek to distance people from Allah's mercy. Allah warns people against this trap in many verses and calls them to ask for repentance and forgiveness:

Say: "My servants, you who have transgressed against yourselves, do not despair of the mercy of Allah. Truly Allah forgives all wrong actions. He is the Ever-Forgiving, the Most Merciful." Turn to your Lord and submit to Him before punishment comes upon you, for then you cannot be helped. Follow the best that has been sent down to you from your Lord before the punishment comes upon you suddenly, when you are not expecting it. (Surat az-Zumar, 53-55)

Allah desires to turn toward you, but those who pursue their lower appetites desire to make you deviate completely. (Surat an-Nisa', 27)

Anyone who does evil or wrongs himself and then asks Allah's forgiveness will find Allah Ever-Forgiving, Most Merciful. (Surat an-Nisa', 110)

Seeing these verses, those who have true faith in the Qur'an can escape the trap of hopelessness.
However, those who insist upon their sins and continue to disobey Allah's commands and prohibitions on the grounds that "Allah forgives in any case" fall into another trap of satan. As long as they do not regret what they have done and turn to the right path, their repentance will have no worth, for Allah makes clear in the following verse what type of repentance He accepts:

Allah only accepts the repentance of those who do evil in ignorance and then quickly repent after doing it. Allah turns toward such people. Allah is All-Knowing, All-Wise. (Surat an-Nisa', 17)

To conclude, those who forsake their former actions and resolutely turn toward Islam are not held responsible for what they did in the past, no matter how sinful their activities were. This assessment also holds true for believers. In other words, believers pay no attention to what people might have done before embracing Islam, for from then on, the Qur'an will regulate their days and they will be held responsible for this in Allah's sight.