A good place to start is an article in your online free encyclopedia, Wikipedia, "Separation of church and state" in world history. Then move on to "Separation of church and state in the United States" (and sources linked to in the following text).
It is ironic that one of several religious groups who fled Europe for the American colonies to escape persecution, the Puritans, then established their Congregational church as the official religion and began their own persecutions. Anglicans imported their version of the established Church of England into the New World.
When the colonies drafted and ratified the Constitution of the United States and its Bill of Rights, however, jealousy and fear between the several Christian denominations resulted in the Establishment and Free Exercise Clauses in the First Amendment, so that the federal government could not make one denomination the official one or prevent any denomination from holding its services.
It is true that several states retained their particular established religion after ratification of the Constitution, along with related measures such as taxing all citizens to pay the salaries of preachers of the established denomination. However, evangelical Baptists persecuted by the established churches adopted the century-earlier call of Rhode Island founder Roger Williams for a "wall of separation between the garden of the church and the wilderness of the world" as the principle to bring about the disestablishment of these state religions. Thomas Jefferson and James Madison became champions of this principle; and disestablishment was largely accomplished in all the states by the early 1800's. For a fascinating description of this alliance, see "The Framers and the Faithful."
It was not disputed in those days that the First Amendment itself was not binding on the states, but only on the federal government. In fact, when then-President Thomas Jefferson wrote that the Establishment Clause erected "a wall of separation between church and state," it was in a letter to the Danbury Baptists, who sought his help in getting the Connecticut legislature to repeal the tax supporting its established Congregational church. Jefferson continued to maintain that "separation" was an adequate description of the intended effect of the First Amendment, as did Madison ("total separation of the church from the state"), primary author of the Bill of Rights.
The first Supreme Court decision to mention "separation of church and state" was a Free Exercise case in 1878 (Reynolds v. U.S.), in which the Court found a Mormon man in the Utah Territory guilty of polygamy, notwithstanding his belief that his religion required the practice. Searching for the meaning of the First Amendment's religion clauses, the Court recited Jefferson's letter to the Danbury Baptists and concluded (in part 5.):
Coming as this does from an acknowledged leader of the advocates of the measure, it may be accepted almost as an authoritative declaration of the scope and effect of the amendment thus secured.
Some twenty years later in an early Establishment Clause case (Bradfield v. Roberts, 1899), the Court held that federal financial assistance to a Roman Catholic-run hospital was permissible because the funding was to an organization sufficiently secular in its charter, purpose, and operation.
After the post-Civil War Fourteenth Amendment was ratified in 1868, including the clause "No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States," the United States Supreme Court ultimately began a gradual process of "incorporation" of portions of the Bill of Rights to apply to state governments through the Due Process Clause.
Most of the early cases "incorporating" the religion clauses were, ironically, Free Exercise cases, brought mainly by Jehovah's Witnesses over local and state government attempts to suppress or regulate their activities: pamphleteering (Lovell v. City of Griffin, GA., 1938), soliciting (Cantwell v. Connecticut, 1940), and refusing to salute the American flag and say the Pledge of Allegiance in school (West Virginia State Board of Education v. Barnette, 1943). The Barnette opinion reversed a previous opinion of the Court which had held that governmental strength should prevail over individual liberties and that complainants' remedy was at the polls. Rather, Justice Jackson wrote:
The very purpose of a Bill of Rights was to withdraw certain subjects from the vicissitudes of political controversy, to place them beyond the reach of majorities and officials and to establish them as legal principles to be applied by the courts. One's right to life, liberty, and property, to free speech, a free press, freedom of worship and assembly, and other fundamental rights may not be submitted to vote; they depend on the outcome of no elections.
Having "incorporated" the Free Exercise Clause against the states, the Court then went on to do the same for the Establishment Clause in 1947 in Everson v. Board of Education, a 5-4 decision upholding state funding for busing parochial school students, even while reciting with approval Jefferson's "separation of church and state" wording.
Another state funding case, Lemon v. Kurtzman (1971) struck down direct aid to Catholic schools for teachers' salaries, textbooks and instructional materials, and formulated the three-pronged Lemon test:
1. The government's action must have a legitimate secular purpose;
2. The government's action must not have the primary effect of either advancing or inhibiting religion; and
3. The government's action must not result in an "excessive entanglement" of the government and religion.
Lemon has been eroded by subsequent cases upholding more indirect aid schemes and finally by a direct aid case, Zelman v. Simmons-Harris in 2002, where a 5-4 majority upheld a school voucher plan, despite the fact that 96% of voucher students used them at religious schools.
Main cases in the public school prayer area have been: Engel v. Vitale (1962) (required recitation of official school prayer unconstitutional); Abington School District v. Schempp (1963) (school-sponsored Bible reading unconstitutional); Wallace v. Jaffree (1985) ("moment of silence" unconstitutional because legislated with religious purpose); and Lee v. Weisman (1992) (graduation prayer unconstitutionally coercive). A related case is Edwards v. Aguillard (1987), holding a Louisiana law -- requiring creationism to be taught in its public schools if evolution is taught -- lacked a true secular purpose and impermissibly endorsed religion by advancing the religious belief that a supernatural being created humankind. Elk Grove Unified School District v. Newdow (2004), the "under God" Pledge of Allegiance case dismissed by the Supreme Court on standing grounds, may soon reappear on the Court's docket with the standing problem cured.
The religious display cases have been close calls and seem to turn on whether the display has a secular purpose. Lynch v. Donnelly (1984) upheld the display of a creche in a park because any benefit to religion was "indirect, remote, and incidental;" whereas County of Allegheny v. ACLU (1989) struck down the display of a creche in the county courthouse, because the "principal or primary effect" of the display was to advance religion. The Supreme Court in 2003 allowed a lower court decision to stand requiring Alabama Chief Justice Roy Moore to remove a Ten Commandments monument from the state supreme court building; but in 2005 by two 5-4 decisions the Court allowed a Ten Commandments display at the Texas state capitol to stay due to the monument's "secular purpose" but rejected displays of the Ten Commandments in several Kentucky county courthouses because they were not clearly integrated with a secular display.
To close this ambitious but long post, here's a link to a very informative webpage, "History of the Separation of Church and State in America," that has many beautiful graphics of early American currency and gives the surprisingly recent history of the substitution of "In God We Trust" in place of our original motto, "E Pluribus Unum," and the addition of "under God" to the Pledge of Allegiance.
The latest iteration of our Supreme Court may indeed be more friendly to "entanglement" of government and religion; but I've tried to show where we are today and how we got there.