Why do you suppose that you think slavery is wrong? How do you know it isn't because you are at the end of a long process of Christianization of morals within western thought, that I (briefly) describe?
I think this is a fair question, and felt it needed some thought, so I took a few days to ponder it. I'm sure there is more to be said on this, but let me outline my thinking process on this.
Where did I come by the concept of why slavery is wrong? I'd say it starts with the way I was brought up, where I learned to treat people well. From there, learning history, I could see that history is filled with examples of people treating others badly (including owning other people as property) and people treating others well. Of course, this part is cultural, and if I had been raised in the South in the 1830's, slavery would not immediately seem obviously wrong. As an adult, one learns about the proper justification for beliefs. How beliefs should be based on evidence, and learning the methods of science, I was able to much better evaluate the reasons behind claims, and also understand the ways in which erroneous justifications have been made for many claims. This ties into psychology, economics, and a whole host of other human activities.
After all this, I've come to accept that morality relates to the suffering and well-being of conscious creatures. As such, we can make objective evaluations about how to increase well-being and decrease suffering. It becomes clear, then, that owning people as property needlessly increases suffering, and as such is objectively wrong. I can then apply this principle in cases where I was raised with a different moral perspective, and find that some of these are wrong. For example, banning gay marriage is wrong (sorry for the double negative!), even if I was raised with the opposite. It needlessly adds to human suffering. So, I can override my cultural teachings on morality, and in the same way I am able to judge other worldviews for obvious cases of immoral behavior (e.g. suicide bombing, genocide, etc...)
How do I know it isn't because I am at the end of a long process of Christianization of morals within western thought? I prefer to rephrase, slightly, saying "why do I believe it is likely..." instead of "how do I know...". This is a matter of probability, and I don't think it is possible to demonstrate the truth of a claim like this to the level of probability that I would consider knowledge. So why do I believe "slavery is wrong" is unlikely to be simply the end of a process of Christianization? First and foremost, I see no clear evidence that Christianity as revealed in the Bible has a clear message against slavery. You can find passages vaguely against particular slave-master situations, and you can find dictates about how you should treat your slaves. There is never a case where it says, clearly, that slavery is in fact wrong. Second, there is historical precedent for Christians to argue for slavery based on a Biblical reading - and this wasn't just a fringe reading. One might argue that they are using a wrong reading, but how do we know?
Thirdly, following on this, my experience with legitimate searches for truth is that they converge after some initial state of disagreement. Take any previously contentious argument in science that is currently well understood - say, the origin of heat. Over the past couple hundred years there were many theories about what heat was - the elemental theory of heat, a fluid theory, a statistical theory, etc... Once the scientific methodology was improved to the point where this problem could be explored rationally, the ideas started to converge. There were fewer interpretations of the evidence considered valid, and now there really is just one. This is true for every branch of science, and I would contend, any approach that addresses the truth - you start off with divergent ideas when you don't fully understand a topic, and converge to a single understanding. I seeno evidence of this in Christianity. There is more disagreement now than 1000 years ago about pivotal doctrines. This is not the pattern we see for something that is true, but is a pattern for something we see a cultural.
Finally, I do see evidence of Christians changing their doctrines to suit their particular ends. Most of the Old Testament is conveniently ignored, interpreted away, because it has become untenable in a modern society. Christian's don't do this because it is a natural, obvious step in the faith but because modern, secular progress has demonstrated that many of the doctrines are either wrong or simply useless. The authority of religion has been battered from all directions.
Thus, I see evidence of cherry-picking, an inconsistent message in the Bible which allows for nearly any philosophical stance to justified with the proper cherry-picking, and a lot of post-hoc argumentation. I see immoral acts not only allowed by God, but commanded by God, and thus at best an inconsistent moral picture from the "good" book. I've heard it referred to as the "great book of multiple choice", and I believe that this interpretation is more consistent with all of the aspects I've seen about Christianity, both present and historical.