From what I have come to understand, the pilgrims came over to what we now know as the USA to flee the persecution they had faced back in the Old World. They brought their beliefs in God with them of course. On the Mayflower the pilgrims had written the Mayflower Compact, which was more of a covenant with Puritan beliefs as how to deal with conflicts and such. From what I have heard the original Mayflower Compact was lost, but the closest we have to it is Mourt's Relation in the 1600's as an account of the event by Edward Winslow and William Bradford and it says:
In the name of God, Amen. We whose names are underwritten, the loyal subjects of our dread sovereign lord King James, by the grace of God, of Great Britain, France, and Ireland King, Defender of the Faith, etc.
Having undertaken, for the glory of God, and advancement of the Christian faith, and honor of our king and country, a voyage to plant the first colony in the northern parts of Virginia, do by these presents solemnly and mutually in the presence of God and one of another, covenant, and combine ourselves together into a civil body politic, for our better ordering and preservation, and furtherance of the ends aforesaid; and by virtue hereof to enact, constitute, and frame such just and equal laws, ordinances, acts, constitutions, offices from time to time, as shall be thought most meet and convenient for the general good of the colony: unto which we promise all due submission and obedience. In witness whereof we have hereunder subscribed our names; Cape Cod, the 11th of November, in the year of the reign of our sovereign lord King James, of England, France and Ireland eighteenth and of Scotland fifty-fourth, Anno Domini 1620
There is also a unique statue that was finished in 1889 called the National Monument to the Forefathers. This monument is in Plymouth, Massachusetts. It was inspired by the pilgrims ideals of a functioning society, which includes the key elements Faith, Morality, Education, Law, and Liberty. It is beautifully done and pretty impressive!
It was actually the US Congress that approved the purchase and distribution of The Holy Bible in schools. Many of these Bibles had a US government stamp inside and are protected books today.
In 1787: Article 3 of the Northwest Ordinance directs the people of the territories to establish schools "to teach religion, morality, and knowledge.” Nearly every state admitted to the Union after this has written in their State Constitution wording that the schools are to teach morality and religion and they all use the Bible as the bases for their teachings.
1920: The ACLU is founded by Alex Baldwin an atheist Marxist. ACLU is cosely connected to Communist Party.
1962: Supreme Court rules that children may not recite a state written prayer in school (Engel v. Vitale). ACLU lawyers will be involved in virtually every case to remove God from American society. This includes removing Ten Commandment Monuments, Crosses, Manger Scenes, Celebrating Christmas in Public Schools, the National Motto "In God We Trust, etc.
1963: An ACLU Lawyer takes the School Prayer case to Supreme Court which bans individual school prayer (Murry v. Curlett). At the same time devotional Bible reading in public schools is outlawed (Abington Township School District v. Schempp).
All in all, I do believe that the United States of America was a Christian nation founded on beliefs in God. Our nation as a general statement has drifted quite a bit from God's laws. Yes, we still have many genuine brothers and sisters in Christ here and we shouldn't forget the many good values that this country once had and steered away from.
I strongly recommend Kirk Cameron's movie, Monumental. It has wonderful insight into this topic.