The primary mentions of religion occur in the Silmarillion. The Númenóreans were apparently of some form of monotheistic faith, only worshipping Eru: (emphasis mine)
Of old the chief city and haven of Númenor 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 Illúvatar, and it was open and unroofed, and no other temple or fane was there in the land of the Númenóreans.
However, later in The Silmarillion, Sauron corrupted the men of Númenor 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 Númenor, 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 city 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 servants 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.
EDIT: In The Unfinished Tales, in A Description of the Island of Númenor, we get further glimpses of the worship of Eru, expanding on what was stated above in The Silmarillion.
Near to the centre of the Mittalmar stood the tall mountain called Meneltarma, Pillar of the Heavens, sacred to the worship of Eru Illúvatar. Though the lower slopes of the mountain were gentle and grass-covered, it grew ever steeper, and towards the summit it could not be scaled; but a winding spiral road was made upon it, beginning at its foot upon the south, and ending below the lip of the summit upon the north. For the summit was somewhat flattened and depressed, and could contain a great multitude; but it remained untouched by hands throughout the history of Númenor. No building, no raised altar, not even a pile of undressed stones, ever stood there; and no likeness of a temple did the Númenóreans posses in all the days of their grace, until the coming of Sauron. There no tool or weapon had ever been borne; and there none might speak any word, save the King only. Thrice only in each year the King spoke, offering prayer for the coming year at the Erukyermë in the first days of spring, praise of Eru Illúvatar at the Erulaitalë in midsummer, and thanksgiving to him at the Eruhantalë at the end of autumn. At these times the King ascended the mountain on foot followed by a great concourse of the people, clad in white and garlanded, but silent. At other times the people were free to climb to the summit alone or in company; but it is said that the silence was so great that even a stranger ignorant of Númenor and all its history, if he were transported thither, would not have dared to speak aloud. No bird ever came there, save only eagles. If anyone approached the summit, at once three eagles would appear and alight upon three rocks near to the western edge; but at the times of the Three Prayers they did not descend, remaining in the sky and hovering above the people. They were called the Witnesses of Manwë, and they were believed to be sent by him from Aman to keep watch upon the holy Mountain and upon all the land.
(Hardback Ed., p. 166)