When Islam dawned on the Arabia, slavery was prevalent not only there but all over the world, Islam didn't invent slavery as many Christian apologists claim but rather uprooted it.

Islam abolished slavery gradually in a way that guaranteed the safety of society against social troubles; in the Arabian Peninsula trade and agriculture depended mainly on the work of slaves. Also some people traded in slaves. So, to avoid a revolution or social insurgences, Islam set rules to eliminate this problem gradually.
These ways could be condensed as follow:
(1) Islam has made freeing slaves an act of righteousness

A Muslim who frees his slave will have a shield against the hell fire. The Prophet (SWS) is reported to have said,

"Whichever man frees a Muslim man, Allah ta'ala will liberate for each of his organ an organ from the Fire."

(2) Punishing a slave severely is atoned by freeing him.

Prophet Muhammad (SWS) is reported to have said,

"Whoever slaps his slave or strikes him, his atonement (kaffara) is to free him."

(3) Relative slaves are to be freed at once.

Islam is emphatic about fostering the relationship between the persons and their relatives.
(4) Killing a believer accidently entails setting a slave free by the sinner to be atoned

Allah (SWT) says in the Glorious Qur'an,

"Never should a believer kill another believer, except by mistake. If anyone kills a believer by mistake he must free one Muslim slave …." [4:92]

(5) Breaking an oath is to be atoned by freeing a slave

Allah (SWT) says,

"Allah does not take you [to task] for what is thoughtless in your oaths, only for your binding oaths: the atonement for breaking an oath is to feed ten poor people with food equivalent to what you would normally give your own families, or to clothe them, or to set free a slave." [5:89]

(6) Whoever says to his wife, ‘you are to me like my mother's back' and then returns to her, his atonement is freeing a slave.
Allah (swt) says:

"Those of you who say such a thing to their wives (i.e. ‘you are to me like my mother's back'), then go back on what they have said, must free a slave before the couple may touch one another again." [58:3]

(7) Whoever invalidates the fast of Ramadan by sexual intercourse should free a slave, this will atone his sin.
It is narrated that someone came to the Prophet (SWS) asking what to do after he had sexual intercourse with his wife in Ramadan, the month of fasting. The Prophet (SWS) ordered him to free a slave.

(8) A slave can free himself by a manumission contract.
A slave can make an agreement with his master of setting him free after paying a certain amount of money. Also the slave holder should help him by giving him money. Allah (SWT) says,

"If any of your slaves wish to pay for their freedom, make a contract with them accordingly if you know they have good in them, and give them some of the wealth Allah has given you." [24:33]

(9) The state treasury is to be spent on freeing slaves
Allah (SWT) says,

"Alms are meant only for the poor, the needy, those who administer them, those whose hearts need winning over, and for (riqab) and helping those in debt, for Allah's cause, and for travellers in need. This is ordained by Allah; Allah is all knowing and wise." [9:60]

His (SWT) statement, "and for (riqab)" means freeing slaves.