If Terah and his family lived in "Ur of the Chaldees" (Genesis 11:26), does that mean he followed the religion of the Chaldeans? Later, they moved to Haran (v. 31). Might that mean that they followed the religion of the Arameans (where Haran was located)? Answers might take account of the interesting fact that Terah's son Abram simply immediately obeyed the Lord when commanded to leave for Canaan (Gen 12:1,4).
We are told that the family lived in “Ur of the Chaldees” (Gen 11:26), which might imply that they followed the beliefs of the Chaldeans—i.e., Semitic people who later joined and became interchangeable with Babylonians. Since from an early age the Babylonians were known to worship a pagan pantheon, the implication would be that Terah and Abram did as well, although Yahweh might have been one of the gods they adhered to.
The family also is to relocate (perhaps back) to the land of Haran (Gen 11:31), suggesting they might have followed the religion of the Arameans. Outside of the Bible, we might note that ruins of an ancient temple to the Mesopotamian moon god, Sin, can be found at Haran.
There are several bits of evidence on this question from other parts of the Bible. In Gen 24, when Abraham’s servant meets Laban, and tells the story giving all the evidence that the meeting with Rebekah was blessed by the Lord, her brother and father repeat back the servant’s frequent endorsement of the Lord: “Then Laban and Bethuel answered and said, The thing proceedeth from the Lord: we cannot speak unto thee bad or good.” (Gen 24:50)
Then we have the story of Rachel, who “had stolen the images that were her father’s” (31:19), i.e., Laban. Since Rachel was Laban’s daughter, she was Nahor’s great-granddaughter; as Nahor was Abram’s brother (Gen 11:26), Rachel was Abram’s great-great-niece. The point, in any event, is that Abram’s family in Haran did worship other gods (probably the curiously-named god Sin).
But later in the same chapter, Laban reaches an important agreement with Jacob, and calls upon God himself to witness it: “The God of Abraham, and the God of Nahor, the God of their father, judge betwixt us.” (Gen 31:53) This makes it sound as there was a shared tradition of worshiping a single God, Elohim, the creator of the universe, which had passed down to Terah, and from him to his sons Abram and Nahor. This might not be correct, however, because the plural verb “judge” (not singular, “judges”), might imply that Laban was referring to a plural number of gods.
Also explicit is this declaration in Joshua: “Terah, the father of Abraham, and the father of Nachor...served other gods.” (Josh 24:2) Clearly, the context shows this means other gods in addition to the only true God, Yahweh.
Finally, the Lord seems to require no special introduction to Abram at the beginning of the next chapter. He simply begins commanding Abram, whose obedience is immediate, or reported next in the narrative; this strongly implies that Abram did already know the Lord.
None of this entails that the worship of Yahweh was entirely forgotten from the days of Noah.
So, on the one hand, we see excellent evidence that Terah’s family was familiar with and did worship the Lord; Yahweh was not entirely forgotten from the days of Noah. But, clearly, the faith had been intermixed with paganism, just as it would be later by the backsliding Israelites.