Q:

What were the beliefs, teachings, and practices of the church regarding slavery in the New World?

When the New World was discovered, slavery was virtually unknown in Europe. Soon after the discovery of the New World, European powers began to make slaves there, and to transport slaves from Africa to the Americas. 
What part did the church (mainly the Roman Catholic church, which dominated Europe) play in this? Did it discourage or encourage the slavery of native peoples?  What were their beliefs and teachings at that time?

A:

Here's what Wikipedia has to say about it (http://en.wikipedia.org/wiki/Catholic_Church_and_slavery#Revival_of_slavery_in_the_Early_Modern_Period) (empasis is mine):
Before Columbus

The Portuguese sought confirmation that they could enslave infidels in a crusade. In 1452 Nicholas V issued the papal bull Dum Diversas to King Alfonso V of Portugal which included the following words: "we grant to you...full and free permission to invade, search out, capture and subjugate the Saracens and pagans and any other unbelievers and enemies of Christ...to reduce their persons into perpetual slavery". In 1454 Pope Nicholas explicitly confirmed the rights granted to King Alfonso V in Dum Diversas in Romanus Pontifex by which he granted to Alfonso "...the rights of conquest and permissions previously granted not only to the territories already acquired but also those that might be acquired in the future".

We [therefore] weighing all and singular the premises with due meditation, and noting that since we had formerly by other letters of ours granted among other things free and ample faculty to the aforesaid King Alfonso – to invade, search out, capture, vanquish, and subdue all Saracens, and other enemies of Christ wheresoever placed, and the kingdoms, dukedoms, principalities, dominions, possessions, and all movable and immovable goods whatsoever held and possessed by them and to reduce their persons to perpetual slavery, and to apply and appropriate to himself and his successors the kingdoms, dukedoms, counties, principalities, dominions, possessions, and goods, and to convert them to his and their use and profit...

In 1456, Pope Calixtus III confirmed these grants to the Kings of Portugal and they were renewed by Pope Sixtus IV in 1481; and finally in 1514 Pope Leo repeated verbatim all these documents and approved, renewed and confirmed them.
These papal bulls came to serve as a justification for the subsequent era of slave trade and European colonialism.

Spanish New World

Slavery was part of the indigenous cultures much before the landfall of the Europeans in America. After the Europeans made landfall in America in 1492, Ferdinand and Isabella saw that, if Spain did not receive from the Pope in regard to the American "Indies" the same authority and permissions which Portugal had received in regard of West Africa, then Spain would be at a disadvantage in making use of her newly discovered territories. Accordingly Pope Alexander VI was approached and already on May 3, 1493 he issued two bulls on the same day in both of which he extended the identical favours, permissions, etc. granted to the Monarchy of Portugal in respect of West Africa to the Monarchy of Spain in respect of America.....and to reduce their persons into perpetual slavery...wherever they may be.
Although the church was excited by the potential for huge numbers of conversions in the New World, the clergy sent there were often horrified by the methods used by the conquerors, and tensions between church and state in the new lands grew rapidly

16th century - Slavery in Europe

Slavery in Europe, mainly around the Mediterranean, continued, and was increased by the increased size of Mediterranean navies to combat the powerful Ottoman navy. The main type of naval ship in the Mediterranean, unlike the Atlantic and Northern seas, was the galley, rowed by galley-slaves; use of the galley only declines from about 1600. The navy of the Papal States was no different from that of Venice, France, Genoa and other naval powers. Galley-slaves were recruited by criminal sentencing, usually for a term of years many never survived, as well as capture in war, mostly of Muslims, and sometimes the African slave-trade. Some of the Popes were personally involved in the purchase and use of galley-slaves.[86] The Ottoman admiral Turgut Reis was captured and made a Genoan galley-slave for nearly four years before being imprisoned and eventually ransomed in 1544. After the battle of Lepanto approximately 12,000 Christian galley slaves were freed from the Turks.[87]
In 1535 Pope Paul III removed the ability of slaves in Rome to claim freedom by reaching the Capitol Hill, although this was restored some years later. He "declared the lawfulness of slave trading and slave holding, including the holding of Christian slaves in Rome".[88]
In 1639 Pope Urban VIII forbade the slavery of the Indians of Brazil, Paraguay, and the West Indies, yet he purchased non-Indian slaves for himself from the Knights of Malta,[89] probably for the Papal galleys. The Knights of Malta attacked pirates and Muslim shipping, and their base became a centre for slave trading, selling captured North Africans and Turks. Malta remained a slave market until well into the late 18th century. It required a thousand slaves to equip merely the galleys of the Order.[90][91]

It appears that the Catholic Church certainly approved of mass slavery and even participated in it. Some clergymen who were sent to those countries were shocked by what they saw and did not really approve of it. But as far I can tell, the Church played a big role in slavery.
You might want to read the complete article on Wikipedia though.

A:

Quakers were one of the first groups who condemned slavery. 
The anti-slavery sentiment started back in the 1600s.  By the 1750s, they were actively trying to have the slavery laws changed in Britain (which ruled Colonial America at that time).
More information
So, clearly there was at least one group of Christians during the Colonial America times who were opposed to slavery.

