The primary mentions of religion occur in the Silmarillion. The Numenorians were apparently of some form of monotheistic faith, only worshipping Eru: (emphasis mine)
Of old the chief city and haven of Numenor was in the midst of its western coasts, and it was called Andunie because it faced the sunset. But in the midst of the land was a mountain tall and steep, and it was named the Meneltarma, the Pullar of Heaven, and upon it was a high place that was hallowed to Eru Iluvatar, and it was open and unroofed, and no other temple or fane was there in the land of the Numenoreans.
However, later in The Silmarillion, Sauron corrupted the men of Numenor to worship Morgoth.
And the first fire upon the altar Sauron kindled with the hewn wood of Nimloth, and it crackled and was consumed; but men marvelled at the reek that went up from it, so that the land lay under a cloud for seven days, until slowly it passed into the west.
Thereafter the fire and smoke went up without ceasing; for the power of Sauron daily increased, and in that temple, with spilling of blood and torment and great wickedness, men made sacrifice to Melkor that he should release them from Death. And most often from among the Faithful they chose their victims; yet never openly on the charge that they would not worship Melkor, the Giver of Freedom, rather was cause sought against them that they hated the King and were his rebels, or that they plotted against their kin, devising lies and poisons. These charges were for the most part false; yet those were bitter days, and hate brings forth hate.
And even in the fall of Numenor, Sauron used the temple to sway those to his side, to ensure their destruction.
Now the lightnings increased and slew men upon the hills, and in the fields, and in the streets of the cityl and a fiery bolt smote the dome of the Temple and shore it asunder, and it was wreathed in flame. But the Temple itself was unshaken, and Sauron stood there upon the pinnacle and defied the lightning and was unharmed; and in that hour men called him a god and did all that he would. When therefore the last portent came they heeded it little.
In the hour that Numenor was overthrown, Elendil was nearly dragged to the Temple to be sacrificed by the servents of Sauron, but he escaped and stood off the coast and watched Numenor fall into the abyss.
In the Lord of the Rings proper, the only mention of temples or religions is a passing mention of the Dunharrow, but no one was sure that its purpose was indeed a temple. TGnat's answer gives an example of what could be considered religious observance in The Lord of the Rings, but any more details to exactly what they were doing is not provided.