If you want the quick version, "separation of church and state" appears nowhere in the Constitution.
It is, in fact, from one of Jeffersons Virginia letters when he was in the process of using the fear of Virginian religious groups to implement a secular system. As far as I remember the fear was that Presbyterians and Methodists would unite and dominate Virginia.
So the main worry back then was, which found its way in the Bill of Rights, that a government would establish a religion and thereby reducing all other religion to the peanut gallery.
So no, it is not in there, but it should be.