I remember learning in my Latin class a few years ago that Rome enslaved the people from countries that the Roman army had taken over. He also mentioned that this was the norm. So why did it eventually become enslaving African natives?
The institution of slavery was a common feature of most pre-modern civilizations. The great exception to this rule was European civilization of the Middle Ages, which, as a rule, only practiced slavery at the margins. That is, after 1100 or so, slavery in Europe was largely confined to areas that were either ruled by Muslims or close to places ruled by Muslims: the Iberian peninsula, southern Italy, and the Balkans.
The chief reason for the elimination of slavery in medieval Europe was the influence of the Catholic Church. This worked in two ways. First, the Catholic Church encouraged secular rulers to promulgate laws against the keeping of slaves and the slave trade. Second, it convinced individual slave holders that freeing slaves was good for their souls. (Thus, a lot of slave holders would free their slaves on their death beds or in their wills.)
The only cases in which the Catholic Church allowed slavery were those in which the slaves were not Christians. In most parts of Europe, the absence of significant numbers of people who were not officially Christian meant that few would-be slave holders were able to avail themselves of this exception. However, in places close to the Muslim world, the presence of both a well-established slave trade and lots of people who were not Christian meant that slavery was able to continue.
The Europeans who were the first to settle in the New World were largely from these borderlands and, in particular, Spain and Portugal. Thus, they were already connected to the slave trading network of the Muslim world, and, in particular, that portion of the network that operated in the western parts of Africa.