The colonists came to America: to build a better economic life for themselves and to have religious freedom. I will focus on the second reason and how it played out in the decision of the Mayflower pilgrims to migrate from England to found the Plymouth Colony in Massachusetts.

James I was the monarch in Great Britain during the period of the Puritan migration to Plymouth. He very much wanted to impose religious orthodoxy in order to legitimize his rule and insisted on adherence to Church of England standards. He tightened up and bore down on dissenting religious groups in a way Elizabeth, on the whole, had not.

As a result of increased persecution that included jailing, fines, and land seizures, the Puritans decided to migrate to Holland, which offered religious freedom. This they did, but realized over time that their children were assimilating to Dutch culture. As they wanted very much to keep to English distinctiveness, they returned to home. Facing intense persecution in England, they decided to migrate to North America. They desired religious freedom. This was not religious freedom as a general principle to be offered to any group but freedom to establish their own form of rigid theocracy. They assumed that in the New World they would be able to obtain both freedom from religious persecution and be able keep their distinctive English culture. In both cases, they were correct.

It is important to keep in mind that the Puritans found remaining an English cultural identity a very high priority. They toiled in 90 degree temperatures in woolen clothes and built English-style house when Native American buildings would have been more practical—not because they didn't realize the problems but because they actively wanted to remain British.

There are numerous reasons colonists chose to settle in America.  The most obvious reason was for an opportunity at a better life.  Many colonists came from Europe, and Europe during the colonial period was quite stagnant in terms of social or even economic growth for the individual.  By coming to settle in America, colonists could expect to find opportunities not present in their native countries, such as jobs, owning land, religious freedom, and even social advancement.  America was the ultimate chance to start new and to create an existence that was entirely crafted by individuals for themselves.  Of course there are a number of other reasons why colonists settled in America, but the reason of opportunity is perhaps the most obvious.

One reason why colonists came to America was to escape religious persecutions.

Those who settled on coming to America did so because they faced intolerant attitudes towards their faith.  Essentially put, the people who came to America from Europe did so because they were being bullied.  Their belief systems were different than what was accepted in England. The society from which the colonists left stressed a homogeneity in religious belief.  Different religious expressions were not seen as socially acceptable.  Nonconformists were harassed for their beliefs. Some of this harassment included imprisonment and, in some cases, even death. 

Those who would not sacrifice their faith set sail for the new world.  For many, undertaking such a difficult journey was a small price to pay for being able to worship the divine in the way they saw fit.  They felt that their attempts would constitute "a city on a hill," a place of sacred worship where religious expression could be fully realized.  Some saw their endeavors as a "holy experiment" of their faith.  Even colonies like Jamestown which were founded for economic motives, still held religious institutions. This shows how spiritual expression was a major part of colonial life.

What was one reason the European settlers came to America?

One of the major reasons that European settlers came to America was to get rich. One should also count the Caribbean settlers as American settlers since they were the first European settlers of the New World. Columbus came to America accidentally, on his way to India in search of spices. The early explorers of America hoped to find a Northwest Passage to reach valuable Asian markets. The Spanish hoped to find the Seven Cities of Gold but instead took Aztec gold back to Madrid. The English settlers of Jamestown hoped to find gold as well; it was not until later were they forced to grow tobacco. The settlers of the Caribbean islands became wealthy sugar planters—no one intended to live in the islands for a long period of time, as the idea was to get back to Europe in order to spend the money in "civilization." The French came to America and found valuable furs which fetched good prices in Europe. The New World's early value was in its wealth of natural resources and the promise of eventually getting to rich Eastern markets.