Does the First Amendment mean there is to be a separation of church and state?
The First Amendment states the following:
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."
The two parts relating to religion mean the following.
1. 'Congress shall make no law respecting an establishment of religion' What this means is that it is unconstitutional to have or promote a national religion. Or a religion that is backed by the government. It doesn't mean that religion and government can't co-exist, but it does mean no endorsements. To teach any one religion in a public school, regardless of how the people may whine, would constitute an endorsement. Or to force prayer to any god.
2. "or prohibiting the free exercise thereof" What this means is that the government can't tell you what to believe, or what not to believe, as far as religion is concerned. Many religious rights activists confuse this idea with that any idea or thought that is contested falls under the same category, but no, just religion. If it is your religion you are welcome to it, and no religions can be banned. However it does not mean that you can do whatever you want whenever you want, say that it is part of your religion and have it be ok. Free exercise, basically includes things that don't affect the ability of others to go about and do as they please. Basic principles of human rights.
These two taken together although they don't forbid church and state interactions, they do make it very difficult for the two to co-exist without it seeming like one or the other is in violation. The current political trend has been to avoid the issue all together with the pure separation of church and state, which eliminates the possibility of either happening.
Supreme Court Interpretation
The US Supreme Court has developed a three-prong test (Lemon-Kurtzman test) it uses to evaluate whether a public activity involves government to the extent of violating the Establishment Clause.
Justice William J. Brennan, Jr. summarized the Court's interpretation of this clause very succinctly in his written opinion for the case Edwards v. Aguillard, (1987):
"The Establishment Clause forbids the enactment of any law 'respecting an establishment of religion.' The Court has applied a three-pronged test to determine whether legislation comports with the Establishment Clause. First, the legislature must have adopted the law with a secular purpose. Second, the statute's principal or primary effect must be one that neither advances nor inhibits religion. Third, the statute must not result in an excessive entanglement of government with religion. Lemon v. Kurtzman, 403 U. S. 602, (1971). State action violates the Establishment Clause if it fails to satisfy any of these prongs."