"The Bible promotes Slavery."
The Bible never promotes slavery but allows for a heavily restricted version of it. Bible approved slavery is the voluntary exchange of labor for material needs—a way for people facing starvation to get food, shelter and security. Everything that makes normal slavery evil is forbidden in the Bible to protect slaves: Kidnapping, forced labor, racism, abuse, exploitation, rape, murder, and owning another person. (Colossians 4:1, Ephesians 6:9, 1 Timothy 1:8-10, Matthew 19:19, Galatians 3:28). If you imagine dehumanizing chattel slavery when you read the word "slave" in the Bible, then you're importing a concept that Scripture strongly opposes. To accurately critique the Bible's stance on slavery you must first understand the slave/master relationship it allows.
RULES FOR SLAVERY IN THE BIBLE
Masters are forbidden from treating their slaves unjustly (Colossians 4:1). This command alone takes away any "biblical justification" a master might think he has for abusing his slave. If an action is unjust, he simply cannot do it to his slave.
Slave and master must treat each other with respect as if they were Jesus himself (Ephesians 6:5-9). Given the importance scripture puts on Jesus, this obviously bans all evils committed against slaves such as abuse, overwork, and treating them like cattle.
Slave and master are equals made in the image of God to be treated with love and respect (Genesis 1:27, Ephesians 6:9). Unjust discrimination against the poor and people of other races is a sin (Galatians 3:28, James 2:1-9)
Both the Old and New Testament condemn kidnapping slaves. 1 Timothy 1:8-10 lists "enslavers" (those who force others into slavery) among the "ungodly and sinful" with liars and murderers. Kidnapping and selling someone was even punishable by death in the Old Testament (Exodus 21:16).
Those who mistreat their slaves will be judged by God for disobeying him (Colossians 3:25).
Most of the alleged "pro slavery verses" are civil laws from the Mosaic Covenant that were never intended for Christians today. These laws were for a specific people (Israel) to govern themselves at a specific time (from the exodus until the the new covenant). The book Hebrews gives a detailed description of how and why the old covenant was replaced.
"Why didn't God just forbid slavery altogether?"
Rather than forbid all of man's flawed economic systems, the Bible forbids what makes those economic systems evil. In the case of slavery, God allowed the voluntary, contractual exchange of labor for material needs and condemns everything immoral and dehumanizing about it. By making this distinction, people facing starvation in societies where slavery was their only option wouldn't have to disobey God just to eat. In the case of ancient Israel, biblical slavery provided the poor with food, shelter, security, protection from abuse, and the dignity of working toward financial independence. In short, God allowed slavery because it's possible to have a slave/master situation that isn't immoral and it gives the poor another option for material provision.