The Bible has many scriptures on slavery and the OT law regulated it to where it was less ungodly than in the surrounding areas. Why didn't the OT law simply abolish slavery ? Probably because it wouldn't have made sense given the culture of those times. People typically sold themselves to slavery because they couldn't sustain themselves otherwise. Or they became slaves because they lost a war, which pretty much worked equally for everyone. Better than being killed or rotting in some dungeon I suppose.
In the OT it was legal for Hebrews to take non-Hebrews as slaves. In the NT there are no regulations on trading slaves, it merely deals with people who already are masters or slaves. Masters aren't told to simply release their slaves but slaves are told to make use of an opportunity to purchase their freedom but aren't told to run away. Paul sent the runaway slave Onesimus back to his Christian master Philemon but told Philemon not to treat him harshly but rather as a Christian brother which is what he had become.
Here's a very peculiar verse which deals with the topic :
1 Tim 6:1 Let as many bondservants as are under the yoke count their own masters worthy of all honor, so that the name of God and His doctrine may not be blasphemed.
It's a bit similar to the modern "as an employee honor your boss and do your work as unto the Lord" and the reason why is among other things that "the name of God and His doctrine may not be blasphemed". Only that these were slaves. I'm sure the Roman empire had regulations on who could become a slave and that abducting free men to make them slaves was prohibited. But some of these people were slaves for life and basically worked for no pay other than food, clothing, shelter. Some of them were treated harshly. Their work was involuntary and for no pay, so the incentive for doing a good job was none.
In our day noone would tell slaves to honor their masters. We would dismiss slavery as an ungodly institution, slave masters as living in sin and we would tell slaves to run away if the chance of succeeding with that was good. And slave masters to simply free their slaves at no charge.
1 Tim 6:1 is a great example of a verse that is very cultural in nature, it deals with an environment where slavery was institutionalized in general and regulated by the government. It was of course different to the system of slavery in the Americas which was based on the abduction of free men in Africa and it wasn't based on racism against any particular group. But for us to apply the verse literally would amount to a horrible misinterpretation of what it is about.