The First Amendment basically states that the US would neither establish nor support a state religion:
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof;"
There are two parts to this statement, both equally important, especially in the context of the late 18th century when it was written. The American colonies were mostly Protestants of varying sects, with a concentration of Catholics (especially in Maryland) and a smattering of Jews, some who had been in North America since the days of New Amsterdam.
European wars of religion between Protestants and Catholics, starting in the 16th century, in addition to the proliferation of Protestant sects, many of whom had been oppressed, targeted, or expelled, made it vital for the budding United States NOT to establish an official church, like the Church of England. However, this in no way indicated that the nation was to be secular. It would have been unthinkable at the time to have had a nation of atheists (or secular humanists). Hence, the second part of the amendment: the new government would not prohibit the free exercise of any religion by its adherents.
So to answer Peter's contention that the nation is secular, the answer is "no", the nation is not secular. The government may not establish a religion nor prevent religious people from practicing their beliefs, but the First Amendment does not create "a secular nation." The government's role is to stay out of the way of religious practice, neither dictating nor preventing it. This way, the government could not exile people, imprison people, or torture and kill them (as had been done in Europe since Martin Luther) based on the existence of a state church or the religion of any given monarch.
Now, does this give freedom from religion? By default, yes it does. Since there is no official state church and the government is out of the membership enforcement business (which is what European government were, in fact, doing), atheists, agnostics, and non church goers could not be coerced by the government into being an Anglican, a Catholic or anything else. This does not mean, however, that the government's job was to enforce atheism--it wasn't. This also does not mean that your God-fearing neighbors can't give you a hard time about not belonging to a church--they can. They just can't deny you the rights guaranteed by your government as a result. So, they cannot prevent you from teaching in a public (state) school, but they can prevent you from teaching in their private religious institution.
As the Bill of Rights is about the rights of the people (and not the rights of the government) it is safe the say that the First Amendment's guidelines on religion were specifically designed to recognize the religious freedoms of the people, not create a government that would enforce secularism.
Of course, there have always been problems between churches and state because certain religious practices impinge on the state's laws: the Mormons and polygamy come to mind, but there have been others. The Amish are a good example, especially in recent decades when the Amish fought (and won) the right to keep their children from attending state high schools and fought for exemptions from certain building codes based on their religious practices. (Amish were jailed for these offenses but were later released.) Great video on the Amish here.
Catholicism has also provided challenges to the First Amendment. "Confessional Privilege" (the right to have confession of a criminal act to a priest be kept confidential and not reported to the police) was gained in an 1813 court case. Such confidentiality was considered as part of religious practice and was protected under the First Amendment. Interestingly, while the Irish Catholics in New York made gains like this one, they actually participated in getting (Protestant) religious instruction removed from public schools in the 1840s:
"Archbishop John Hughes is born in Ireland, where he witnesses oppression of Catholics by the country's ruling Protestant minority. After emigrating to America, he is ordained in Philadelphia and then moves to New York, where parents have taken many of the city's 12,000 Catholic children out of the public school system. They see the schools as bigoted against Catholics and object to the use of the Protestant King James version of the Bible. Arguing that no religion should be favored above another, Hughes petitions the city council, demanding Catholics be given money to set up their own schools. After losing the vote, he turns to politics, urging Catholics to vote for his slate of candidates in the 1841 state elections. Nearly all of his candidates win, and in 1842 the state passes a bill ending religious instruction in public schools. Four days later, riots break out; bricks are thrown through Hughes' windows, and the doors of his house are kicked in."
So the impetus of the secularization of public schools actually begins with the Catholic church. Go figure. Of course, you're really talking about an Irish vs English political rift underneath the religious argument as well. The Irish were not too welcome in the US during their potato famine, brought on, of course, by overfarming and the plantation system that the British had brought to the Emerald Isle.
The other problem, of course, is what happens when the state strikes down its own laws related to moral behavior, which are supported by many religious Americans. For example, sodomy laws were consistent in almost every US state up to the 1970s. The state's law coincided with religious "law" (or a religious concept of "sin"). Of course the state's interest in these laws was not a religious one, although the interest of the general population may certainly have been. The state was interested in the social stability gained by (heterosexual) marriage, which was the best guarantee of security for children. As has been mentioned elsewhere, the family preceded (and superseded) the American Constitution and was regarded as the building block of societal stability. Homosexual behavior was seen as undermining the whole paradigm of the family and, therefore, of stability. While dovetailing with religion, the state's interest was not construed as a religious one.
However, the gay rights movement based its argument on the state's interest in applying all its laws equally to all citizens (the "Equal Protections" clause of the 14th amendment, originally designed for freed slaves). This legal argument--which preceded the cultural shift we have seen over the past 20 years--was the basis for the legal challenges to the sodomy laws. As sodomy laws fell and homosexuality was no longer criminal, this opened the door to open homosexuality and the demand for not only rights but representation. To religious people, this is a major problem, as homosexuality is still a sin in many religious circles. While not necessarily opposed to decriminalization, many religious people ARE opposed to its current moral neutrality and, here in California, to its active promotion in the public schools. Suddenly, something your religion teaches you is a terrible sin is now a behavior protected (and in CA promoted) by the state. The state and religious Americans are now at odds. This is what brings out the political activism that you are seeing. There is no established church to prevent this, and the government has been argued out of sodomy laws, so it's up to the religious to prevent what they perceive as a sin from becoming legitimized and promoted.
Oddly enough, the gay rights movement has turned to marriage as its way to gain acceptance. The very building block of society that homosexual behavior was seen to undermine is now the brass ring that the movement is fighting for. In some ways, this is gratifying since there is a desire for societal stability. In other ways, especially if you read Michael Savage, it's rather threatening, since Savage actually wants to change the nature of marriage once gays obtain the right.
However, and this goes back to Peter, the religious in this country have as much a right to fight what they consider to be wrong, sinful or evil behavior as the gays have a right to fight what they consider to be oppression. The First amendment secures the right for the religious to practice, freed of state interference, and the history of this country demonstrates over and over again how the religious folk in America fight to make their notion of right and wrong the law of the land. Whether you agree or not is immaterial: the right to the free exercise of religion is guaranteed and, sometimes, that free exercise will have political ramifications. It's inevitable.