When and how did people begin to consider slavery immoral?

Let's not forget that you could sell yourself into slavery (and keep the money of course, maybe a fee to slave traders if you needed them).

This was a COMMON practice in Roman times. Specially educated Greeks, whose homeland had been devastated by never ending conflicts, would usually sell themselves into slavery to Roman masters, where though they became property, and could be sold, they got wages and food and shelter and could teach and form families, a much preferable outcome than starving or dying in a war in Greece.

Wikipedia states:

Essenes
Slave ownership was widely accepted by the majority of early Jewish societies, but the Essenes were a small, ascetic sect that reportedly renounced slavery,[16] although some scholars question whether the Essenes actually renounced slavery.[17][18]

The sources for those references is Hezser, Catherine, "Jewish slavery in antiquity", Oxford University Press, 2005

Please note that due to economic considerations, non-slavery society was not a very viable proposition in ancient times.

As Mark C. Wallace very correctly points out, the British antislavery movement of the 18th and 19th centuries was the first serious anti-slavery movement that managed to roll back slavery. However, no discussion of anti-slavery can be complete without a mention of the good friar Las Casas.

Sublimis Deus a Papal encyclical promulgated by Pope Paul III in 1537 condemns slavery and declares it null. Native population of South America is declared to be rational beings with soul.

Nobody so far has pointed out the obvious:

The first people to believe that slavery was evil and wrong were probably discontented slaves (as opposed to contented slaves).

I would guess that nobody was as likely to notice and disapprove of his boss's unjust treatment of him as a slave would be.

In ancient Rome the First Servile War was in Sicily from 135 to 132 BC and involved about 70,000 to 200,000 rebel slaves. The Second Servile War in Sicily lasted from 104 to 100 BC. The Third Servile War was on the mainland of Italy from 73 to 71 BC and involved 120,000 slaves led by Spartacus. These rebel slaves were mostly interested in gaining their own freedom and the freedom of their friends and families. But it is quite possible they sometimes expressed the opinion that all slavery was wrong, and that other people heard of that opinion.

The Zanji Rebellion in southern Iraq from 869 to 883 was one of the biggest and bloodiest revolts against the the Abbasid Caliphate and the rebels included black slaves and free men. It is certainly possible that some of the slaves involved expressed the opinion that all slavery was wrong.

And there have been many other revolts of slaves who mostly sought their own personal freedom, but may have sometimes said that all slavery was wrong.

The Institutes of Emperor Justinian (ruled 527 to 565) say:

. . the law of nations is common to the whole human race; for nations have settled certain things for themselves as occasion and the necessities of human life required. For instance, wars arose, and then followed captivity and slavery, which are contrary to the law of nature; for by the law of nature all men from the beginning were born free.

and:

Slavery is an institution of the law of nations, against nature, subjecting one man to the dominion of another.

And these can be interpreted as anti slavery opinions. Certainly many 19th century southern Americans expressed opinions far more favorable to slavery.

An early American protest against slavery was the Germantown (Pennsylvania) petition in 1688 by four German Quakers and Mennonites.

Slavery has been abolished by many Chinese governments for various reasons, the first time by the Qin Dynasty of 221 to 205 BC.

The Sachsenspiegel lawbook of about 1220 condemned slavery as a violation of God's likeness in man.

King Louis X of France decreed in 1315 that "France means freedom" and any slave entering France was to be freed.

And so on. There are many possible anti-slavery expressions throughout history, making a definite history of opposition to slavery difficult to wwrite.

It is incorrect to perceive that there was a single concept of slavery in ancient world. The Latin word for slave, "servus" at the same time meant a servant.

The concept of slavery differed very much between ancient societies and also differed in time. Sometimes a slave would be considered a member of the family to the extent that a formal kinly relationship such as formal adoption would be required for having a slave while in other societies the concept of slavery rose from the captured enemies or even possibly, rebels. Still in others one could become a slave for failing to pay a debt and could earn freedom back upon paying the sum or serving a certain term.

The societies differed much in the social status of "slaves" and their rights and the very word for slave could only barely be translated from one language to another without loosing a part of the meaning.

We currently apply the same term "slave" to penal farm convicts, prisoners of war, debtors forced to work to repay debts, dependent peasants, minor clan members and so on, who all had different legal status.

The situation is quite similar to how we apply the term "king" to various ancient titles that had little in common (rex, basilios, vanax, archon, archagestes, despotes, hegemon, tyrannos etc). There was little in common between elected and not heriditary Roman rex responsive to the senate and a Egyptian pharaon yet we call the both "kings". If a Roman of the time appeared in modern day he would possibly call the president "rex" and the penal farm prisoners "public slaves".