This part will emphasize the less well-known religious social foundations of America.
It is important to note the adjective 'social' as many today seem to only think in political terms. In the early modern period, before the rise of large, integrated, bureaucratic states, politics was only one of many aspects of a nation.
The social aspect, the institutional structures of family, school, church, government, etc., is the formal organization of the underlining cultural organism. The culture is the local, private and semi-private expectations and worldview outlooks that affect society. Naturally, there is a reciprocal relationship, but usually the larger institutions (such as the government) reflect the beliefs of the culture as a whole.
Jamestown, Plymouth and Massachusetts Bay
In both aspects of early America, religion played a dominate role. The wide-spread localism of this period allowed for religious and social diversity within a Christian context. Naturally, the localism arose from the vast size of the Eastern coast. Even so, Protestantism tied these diverse settlements together.
In 1607, Jamestown, although starting as a business venture of the Virginia Company of London, included a minister. And worship services were required morning and evening every Sunday. Catechizing the young came a few years later after women showed up. The particular denomination was Anglicanism. And its 39 Articles were clearly Protestant with a strong strand of Reformed thinking (the sovereignty of God and the depravity of man, here).
Presumably, many Americans know that both Plymouth and Massachusetts Bay were founded by Protestants: Separatists and Puritans respectively. Both groups were ardent Calvinists. And they came for religious freedom.
Culture & Christianity
Parents were to inculcate their children with Christian practice and doctrine. That included especially the Bible and the catechisms. Church leadership especially encouraged this in the families all the while they catechized the same families and their children. The schools simply reinforced this Protestant outlook with Bible readings and the Puritan New England Primer.
Although church membership was low (probably due to the high admittance standards), attendance was over 50% through the 1700s. Virtually all Americans were Christians of one stripe or another.
From Bibles, catechisms and sermons, most of the books were religious in nature. One of the most popular children books for over 100 years was a Puritan poem about judgment day, the Day of Doom. Newspapers, speeches and debates were couched in religious language, especially the Calvinist language of "providence." Even Paine's Common Sense used Christian language and imagery.
Politics & Christianity
Election day sermons were the mainstay in New England, while practiced occasionally elsewhere. This old tradition gathered the state leadership into one building to hear the chosen minister expound their duty to God. Several such sermons included a public defense of resistance to tyrants. Sermons were also preached during artillery drills, funerals and public holidays.
Political leaders, one and all, spoke the language of Christianity. Many were devout Protestants (John Jay, Patrick Henry, Roger Sherman). A few may have been borderline Deists (Washington). And even fewer were outright Deists (Jefferson). And some were hard to figure out (Madison).
Yet the Deism of Jefferson was not publicly known. And the Christian climate of the time was such that the stigma of the title 'deist' was even avoided by Jefferson. During his run for president in 1800, he was accused as such (without any real evidence). He publicly denied the charge.
The Declaration of Independence (as the organic foundation of America) explicitly mentions God and providence, rooting American liberties in Christianity. The Continental Congress pronounced several days of prayer and thanksgiving in explicitly Christian language, enacted public prayer and implemented chaplains.
All those state constitutions mention God and religion explicitly. The lack thereof in the Constitution makes sense in light of the state and local concerns of a nation-wide establishment of a single Christian denomination--what mother England had at the time.
Nevertheless, the new Congress still funded chaplains, asked for days of thanksgiving (via Washington), attended public facilities for worship services, and even condoned an American edition of the Bible (more here).
Several state constitutions still had a form of Christian establishment after the formation of the Constitution, with some including religious vows. In fact, the 1778 South Carolina constitution stated:
"The Christian Protestant religion shall be deemed, and is hereby constituted and declared to be, the established religion of this State."