Let me deal with the part about religious freedom here first. The claim you have heard is true. The Pilgrims at Plymouth and the Puritans who founded the Massachusetts Bay Colony wanted to practice their own faith. They did not support any wider concept of religious freedom. Other religious groups were barred from the colony and they had laws against blasphemy that were used to protect their version of theology

Baptists were persecuted in the Massachusetts Bay Colony during the 1640s, and explictly banned by law in 1645. Punishments could be harsh. In 1651 for instance a group of three Baptists were arrested in Lynn, Massachusetts and convicted of re-baptizing adults (Baptists did this because they did not believe infant baptism was valid). All three were fined, one was whipped.

Quaker fared worse than Baptists. In the 1650s Quakers who visited Massachusetts were branded, whipped or mutilated. Puritans who converted to Quakerism had their property seized. In 1659 and 1660 Massachusetts executed four Quakers; Marmaduke Stephenson, William Robinson, Mary Dyer, and William Leddra. The Puritans would have likely killed more Quakers, but they were commanded to stop by royal command in 1661.

Puritans also tried to purge dissenting theology within their own ranks. Puritan Minister Roger Williams claimed that the Puritans had illegitimately occupied native lands, which resulted in him getting tried for sedition and heresy. He was banished from the colony in 1635 and ultimately founded what would become the Colony of Rhode Island. Rhode Island allowed for religious toleration.

In 1638 the Puritans tried Anne Hutchinson in what is now called the Antinomian Controversy. The theology here is a bit complicated, but it resulted in the Massachusetts expelling Hutchinson and a number of other dissenters from their churches and banishing them from the colony.

The Puritans were more repressive than England, but it was matter of degrees. In England it was not until the 1689 Toleration Act that dissenting Protestants, who were not members of the Church of England, could worship freely. This provided religious toleration to Baptist, Quakers and Puritans, but did not allow such freedom to Roman Catholics, atheists, or non-Christians. That said, how much repression dissenters actually faced in England could vary a lot, both Quakers and Baptists still spread widely during the seventeenth century. Quakers and Baptists were sometimes arrested, but they were not killed like they were in Puritan New England.

The idea that society should tolerate of many religious views is a comparatively new one, it was only becoming common in Europe at the end of the seventeenth century. In the American colonies only Pennsylvania and Rhode Island would allow significant religious freedom for their residents. The Puritans were harsher than the England in enforcing their state religion, but both groups made religious dissent a crime.

To your first question: anyone in England would be expected to belong to the Anglican Church, to conform to its rules and its dogma in much the same way the Catholic Church expected conformity. This conformity was controlled by a hierarchy of priests bishops and archbishops, with the Archbishop of Canterbury at the head. People were expected to go pay tithes to the local church, attend the services of the appointed prelate, curate, or priest, go through the prescribed rites of birth, marriage and burial. This was a matter of law: the Uniformity Act of 1559 specified fines and punishments for people who did not attend or preached outside of Church authority. There was more repression under the Seditious Sectaries Act of 1593.

When the Brownists ( AKA Pilgrims) separated themselves from the Anglican church and gathered to have church services among themselves, in one way they were more of a threat- many of the Puritans , as the name implies, wanted to stay inside the Anglican Church and, as USReligionScholar says, make it less Catholic: the concept of a human institution being able to grant salvation was distasteful to them. But to the Brownists ( though they would have changes of opinion, and Brown himself would recant) it was wrong to actually consort , be in a church, with people who were in error and, likely, damned. They created their own congregations, gathering under no authority but the general will of the congregation. John Whitgift, the Archbishop of Canterbury under Elizabeth I, went to great lengths to try to stamp them out, and his successor, John Bancroft, continued the repression. Their church services were regularly raided, their members- especially their preachers- were often imprisoned and, in a few cases ( like Henry Barrow and John Greenwood), executed.

On an informal level, there was undoubtedly far more trouble for them: this was a time , as USReligionScholar says,in which uniformity of belief was expected and non-uniformity was deeply unsettling. Even when the Brownists were not being arrested by the authorities, from William Bradford's writings we can infer that they were constantly harassed by locals, their businesses shunned, their houses continually watched for signs a forbidden religious service was being held...and, when they came to Plymoth, they and the Boston Bay Colony would carry on the same attitude and be equally intolerant of dissent, if not more so.

During the chaos and disruption of the Interregnum of the 1650's, after the English Civil War, enforcing a uniformity of belief became impossible, and a great number of sects appeared. After the Restoration in 1660, there would be only limited attempts to harass them. But still: taxes, tithes, would be levied on everyone to pay for a local Anglican church, and even into the 19th c. non-conformists and Catholics would be in theory barred from going to a university like Oxford or Cambridge.