American schoolchildren learn that the Europeans who came to settle Plymouth colony did so for religious freedom. The implications of this story is that religious freedom and the separation of church and state came about as a natural consequence of these settlers' original motivation. The link between Plymouth and religious freedom was invented in the 19th century as the first historians of the new United States--especially the prolific New Englander George Bancroft--worked to create an American history that explained the country's values. At the same time Plymouth Rock--and the paintings ludicrously depicting new arrivals stepping out on to a rock, when every sailor knows partially submerged rocks are to be avoided--became part of this invented past.
In fact, most migrants who came for religious reasons did so for freedom to pursue their own religion but without any particular attachment to the idea that other religions should be protected. Freedom of religion in general--which the United States upholds--was not valued. Rather the Plymouth settlers hoped to be left alone to do what they thought best, without interference. First they fled England for the Netherlands. Subsequently they left the Netherlands, enticed by the opportunity to live independently and separately, in what was then the far north of Virginia. Their immediate goal was separation from other people with other religious views. (For reasons related to their rejection of the Church of England they were labeled "separatists" but that term had a different resonance.)
What the Plymouth migrants learned was that pursuing their religious vision in America was not a straight-forward proposition. Other people with other views came along after them, confronting them with the need to decide whether to mete out the same persecution they had once endured. (To their credit, the people in Plymouth generally decided not to do so.) Eventually the sheer number of migrants there and elsewhere in colonial America who wanted religious freedom for themselves created conditions amenable to a principled commitment to freedom of religion for others (although this was the work of some generations).
A number of factors went into the shift. The Plymouth settlers were not the only ones to come to pursue their own religious truth, so that by the time of the American Revolution the thirteen colonies encompassed a huge variety of different faiths, most of them some form of Christianity. No single Christian church dominated, many of them were at odds one with the other, and a sprinkling of Jews and Muslims added to the diverse mix. Creating a nation out of that diversity appeared a quixotic aim. Promising each person freedom to practice his or her own faith assuaged concerns that joining the new nation would expose them to religious persecution. The relationship between the Plymouth founders' religious views and the later disestablishment of religion was at best complicated and indirect.
So, pretty much everything you learned about Plymouth was slightly off: they did not call themselves "Pilgrims," they did not land on a rock, and they did not intend to establish religious liberty. Neither did they eat much turkey, another nineteenth-century addition to the national origins story.