DID THE PILGRIMS COME TO AMERICA FOR RELIGIOUS FREEDOM?
Not in the sense that you are probably using that phrase
When we think of religious freedom, especially in our pluralistic culture, we usually mean the right of all individuals to practice their religious beliefs as they see fit. This was most certainly NOT a principle to which the Pilgrims were committed.
The Pilgrims believed they were rightly practicing Christianity and that the established state churches in Europe were not. However, the state churches refused to allow the Pilgrims to practice their faith outside their institutions, so the Pilgrims fled in response to the ensuing persecution.
So they certainly fled to America for religious freedom for themselves. However, they could be quite harsh in dealing with religious dissidents within their own communities. What the Pilgrims valued was establishing right-practicing Christian communities ... NOT religious communities where individuals were free to practice their faith according to their own conscience.
What? You're not a Calvinist? You know what that makes you, don't you?
DID THE PILGRIMS COME TO AMERICA TO SPREAD THE GOSPEL?
Ummm, sort of?
The Pilgrims came to America to consciously create a "city on a hill." However, I would say that the "light" from this "city" was intended to shine more brightly on Europe than the Americas. They hoped that by establishing communities that rightly practiced Christianity they could bring about conviction and repentance in Europe through their example.
That said, the Pilgrims were interested in evangelizing the Native American population, at least in an official sense. In fact the original seal of the Massachusetts colony had a picture of a Native American saying, "Come over and help us." (This is a reference to a story in the book of Acts in the Bible where Paul receives a vision of a man from Macedonia asking him to come share the gospel with them.) However, actual evangelistic efforts by Pilgrims among Native Americans were quite limited.
I would say that the Pilgrims saw themselves as establishing right-practicing Christian communities with more of an eye toward church reformation in the Old World than evangelism in the New World, though the latter was certainly not completely absent from their plans.
Hey Europe! Look over here!