You might like to see the discussion in the FAQ at Was America founded as a Christian nation or a nation with Christian morals?.
But that's not to discourage any further discussion or any more questions that you might have.
So this is actually something that I discuss in detail on my blog in two seperate, but linked posts that you can find here and here and I will mostly be pulling my answer from there.
Short Answer
While the vast majority of America identified as a denomination of Christianity, they did not see themselves as a unified "Christian nation" or even as a united "Christian religion."
The General Public
There were many different denominations of Christianity that existed during those early years of Americas history, and as we will see, many of them distrusted or disliked one another. Some faiths actually tried to demonize other branches of Christianity as well. Because of this, the idea of talking about America being founded as a Christian Nation is tricky because all Christians didn't see themselves as one unified faith, but rather thought that some sects had it right and others were wrong. Still, religion was important to many Americans, just the level of that impact on their lives varied on many factors
Over a dozen sects of Protestant Christianity was in place in the American Colonies at the time of our founding, and may had deep animosity for one another. Anglicans hated Catholics and Methodists, Methodists hated Quakers and so on. It's also worth noting that practicing one's religion (including attending a church service at least once a year), was much lower than most contemporary Americans might expect. Less than 20% of Americans attended a church service annually by 1775, with some parts being dramatically less (in rural New York, where present day Vermont is, the number was around 10% church attendance in a given year). However, in some areas, church attendance was much more frequent (suchas South Carolina where they had rates of around 30% church attendance per year). Still, it's also worth mentioning that since the Second Great Awakening occurred, (around 1730-1750) new sects of Christianity had popped up, so religious diversity is something that did flourish, but again, it wasn't viewed and practiced the way we think of Christianity today.
The Founders
Many of the Founders viewed Christianity in completely different ways then compared to the general public. Part of this is due to their education, with many of the Founders having spent years and even decades studying theology and the history of the New Testament due movements in the field of higher thinkings. These movements in Christianity were largely a response to the Enlightenment and, later on, modernity. However, the Enlightenment is what primarily shaped the Founders as they largely operated before the peak influence of these later Christian movements.
You have some Founders, like Jefferson, whom most people would call a Deist (although he would insist he was one of the few real Christians), and had revolutionary views of Christianity. Jefferson was best described as a deist who believed in the teachings of the New Testament, but not in its miracles or supernatural elements. That said, Jefferson also claimed, "I am a real Christian, that is to say a disciple of the doctrines of Jesus," (referring to the moral practices of Jesus). He dismissed parts like the virgin birth, equating it to similar stories found in Roman mythology (which he also studied). We further see the influence the Enlightenment had on him in his critique of pre-Enlightenment Christian theologians. For instance, he said:
"The metaphysical abstractions of Athanasius and the maniacal ravings of Calvin, tinctured plentifully with the foggy dreams of Plato, have so loaded [Christianity] with absurdities and incomprehensibilities... that it had lost its native simplicity and purity."
To Jefferson, Christianity provided a moral compass for people to model their lives after. However, as the above quote alludes to, Jefferson believed that the early Christian authors along with more contemporary theologians had distorted the bible and led people astray. In an 1814 letter to his friend, John Adams, Jefferson even wrote:
To try and help fix these biblical problems (from his perspective), he created his own bible known as the Jefferson Bible, which essentially was a compilation of the Gospels that left out all of the miracles and supernatural occurrences. Unfortunately for him, his deistic views of Christianity did hurt him politically when in the 1790s and 1800s his opponents denounced his beliefs in newspapers, with some even labeling him the anti-Christ.
Regardless, religion was deeply important to Jefferson throughout his life. He authored the Statute of Virginia for Religious Freedom Bill in the late 1770s, which protected people from having their rights trampled by religious institutions or religious governments. Protecting other deists, or even agnostics and atheists, was a priority for Jefferson in helping the bill get passed. When he passed away, it was one of only three accomplishments he wanted to be attributed to him on his headstone (even over his roles as Governor of Virginia, Secretary of State, Vice President, and even President).
On the flip side, some Founders, like Ben Franklin, were much more indifferent to the concept of Christianity. Franklin was of course a notable atheist – the most notable of the bunch who cared little for organized religion. Like Jefferson, he did revere the teachings of Jesus but remained skeptical of the faith, saying:
As to Jesus of Nazareth, my Opinion of whom you particularly desire, I think the System of Morals and his Religion as he left them to us, the best the World ever saw, or is likely to see; but I apprehend it has received various corrupting Changes, and I have with most of the present Dissenters in England, some Doubts as to his Divinity; tho' it is a Question I do not dogmatise upon, having never studied it, and think it needless to busy myself with it now, when I expect soon an Opportunity of knowing the Truth with less Trouble.
To paraphrase, Franklin never saw Christianity as something worth studying, and since he was near the end of his life, he’d soon be dead and find out once and for all if any of it was true.
But of course, it is worth mentioning that many Founders were traditional types of Christians -- where as they believed in the teachings of the New Testament and the divinity of Jesus. One such example is John Adams. Adams was a Unitarian and was a church-goer himself. He and his wife often prayed together and his letters frequently finished with lines such as this one found in a letter to his wife in 1776, “I am, with constant Wishes and Prayers for your Health, and Prosperity, forever yours.” Indeed, for Adams, his faith was an important part of his life that he shared regularly in conversation with others. However, it is important to note that he did not let his religious beliefs impact his political ones. Furthermore, he was quite skeptical of American religion and wrote about this, saying things like:
This type of religious understanding from Adams is important because even though he appeared to hold deep religious convictions, he saw it as an insult to allow those convictions to influence his politics.
In the end, talking about religion during this era is complicated. This goes double if we include the Founders in that discussion. Here, I gave examples of people with faiths all across the board -- from true believers like Adams to atheists like Franklin and many points in between. The real take away, in my opinion, is that the majority of Americans identified as Christian, but very few practiced their faith in a regular and organized fashion as many think of religion today. Likewise, the Founders didn’t see religion in the same light many see it today, and even those that did saw the creation of the new republic as a secular event. Our founding documents (Declaration of Independence, Articles of Confederation, Constitution, Bill of Rights) are overwhelmingly secular, and much of what they speak about can be found within the pages of other secular documents of their day (like the Magna Carta or the English Bill of Rights, 1689). Even more notable is the fact that the many journals, letters, and notes taken from the delegates at the first Constitutional Convention clearly showed their desire to create a separation between Church and State, since the British, who had hindered their freedoms, had tried to link the Anglican Church to the colonial governments. So was America founded as a Christian nation? Well, that’s for you to decide.