What role did religion play in creating the colonies in the New World?

Well, you can start with the Pilgrims, a Protestant group who settled Massachusetts and was one of the earliest settlements, (though not the first,that was Jamestown, Virginia, which was not religious). Then you probably want to include the Amish and the Shakers of the Midwest.

Well there is also the escape from British religious persecution. The colonists wanted to freely decide what faith they wanted to practice instead of being threatened by law to accept a faith by threat of imprisonment or death. Not that it really solved it anyways since we still had religious persecution in the colonies later. But it helped establish secular values to accept differences in religious beliefs. That was the goal and for the most part it has worked even though to this day denominations battle to make their church the main US denomination. Scary day that will be if they succeed.

I think religion did pla a big role. Not only because of reliogious reasons, but when in a foreign country your language and your church keeps you together.
The church was the meeting place for news, pastors came first from the homeland and later educated in USA. Colleges were built. We have St.Olaf - Norwegian, Gustavus Adolphus - Swedish and Dana College.
Eric Norelius, Vasa, Godhue County in Minnesota was important representativ for the Lutheran Church in USA.
Erik Jansson and a group of Swedish immigrants seeking religious freedom left Sweden The site for their new utopia, later called Bishop Hill, These Janssonists built a successful communal society based on economic and religious principles. Many of the structures erected during this building
campaign still exist.
Every country has its story and can fill more than three pages. You only need to choose one country to represent all the others.

The so-called Pilgrims were not a religiously homogeneous group--some were Puritans, some were dissenters and a few were even Catholics. When in 1630 a large group if Puritans arrived with John Winthrop in the Massachusetts Bay colony, they certainly did not deserve to be called religiously tolerant. They drove out anyone who did not display their brand of orthodoxy. Roger Williams and Anne Hutchinson both fled Massachusetts, and Williams formed the Rhode Island colony in response. Lord Baltimore and the Calvert family were given the colony of Maryland, as a haven for English Catholics. Nevertheless, most of those who governed the colony were Anglicans--that being thought to be more politic in dealing with the home country.

In 1681, Charles II granted a huge swathe of land ostensibly running from the Atlantic to the Pacific Ocean--they were not really strong in geography in those days. It was in payment of a debt to Admiral William Penn, and was made to his son, William Penn. Pennsylvania was the only colony which truly did practice religious tolerance.

EDIT: During the Second Angl0-Dutch War, James, Duke of York, took the Dutch colonial city of New Amsterdam. It was renamed New York in his honor. The Dutch had earlier taken the Swedish colony in the south of their territory. When it was divided into three colonies--New York, New Jersey and Delaware--the Dutch and Swedish settlers in place were allowed to practice their religions unmolested. That was Dutch Reform and Lutheran. They were more or less religiously tolerant by default, although that didn't run to tolerating Catholics.