One of the more common reasons given to school kids for the Pilgrims leaving England was so they could establish religious freedom. Well, that's only partially correct. Religious freedom to the early Puritans was meant only for their religion.
In order to understand the theology of the Puritans, a brief history lesson is in order.
For decades up until 1533, Protestantism had been the main religion of England. In that year, however, Mary Tudor (Catholic Queen Mary) succeeded Edward VI. To escape persecution from the Catholic church, the Protestants beat feet and fled to the European continent.
The story doesn't end there. Due to all the internal strife in the Lutheran areas, the exiles weren't welcomed and, in some cases, were turned away. On the other hand the Reformed Calvin areas welcomed these exiles with open arms.
Why is this important?
Simply put, Lutherans believed in a "two-Kingdom" theology in which there was some modicum of separation between the Church and the State. Reformed Calvinists, however, believed in a "single Kingdom" theology in which the State and Church were inseparable and BOTH were subject to the dictates of God, revealed through scripture.
When Mary died in 1558 and was succeeded by Elizabeth, the exiles returned home. Strengthened by their Reformed Calvin education, they looked to "reform" the English church and make Reform Calvinism the primary influence in English life.
When it was inevitable their goal wouldn't materialize, the separatists left England. They truly believed they were the "elect", just as the Israelites of the Old Testament were, and leaving for the New World was the modern version of the Exodus.
Arriving in America, the Puritans sought to establish a purely Christian commonwealth. This was further codified in 1641 in the Massachusetts Body of Liberties. If one has never read this document, it truly is an enlightening read. In it you'll find exemptions for when it's ok to torture confessions from people, and a list of Capital crimes straight out of the Pentateuch.
Any person who settled and disagreed with the Puritan ethic was typically tried and convicted of blasphemy and either killed or exiled (see Roger Williams and Anne Hutchinson).
In essence, two things can be gleaned from this:
1) "Religious freedom" was granted as long as that religion was Reformed Calvinism.
2) America's Christian founding, of which the MA Bay Colony can be considered the first permanent settlement, was intended to be a Christian theocracy of the Reformed Calvinist tradition.