What were the religious activities during the time of our patriarch, Abraham and his father, Terah? Did all know of God or did God call Abraham uniquely?

We can be certain that Noah and his sons knew the true God and the true manner of worship. According to the biblical chronology (in Genesis 11), there were 222 years from the time of the flood until the birth of Terah, the father of Abraham (originally, Abram). Terah was Noah's seven-greats grandson. Terah was 70 years old when he began to bear sons (Genesis 11:26And Terah lived seventy years, and begat Abram, Nahor, and Haran.
See All...). However, a comparison of scripture with scripture (Genesis 11:32And the days of Terah were two hundred and five years: and Terah died in Haran.
See All...; 12:4
See All...; Acts 7:2-4 [2] And he said, Men, brethren, and fathers, hearken; The God of glory appeared unto our father Abraham, when he was in Mesopotamia, before he dwelt in Charran, [3] And said unto him, Get thee out of thy country, and from thy kindred, and come into the land which I shall shew thee. [4] Then came he out of the land of the Chaldaeans, and dwelt in Charran: and from thence, when his father was dead, he removed him into this land, wherein ye now dwell.
See All...), shows that Terah was probably 130 years old when Abraham himself was born. That would make the birth of Abraham 352 years after the flood. It might seem that this was a period that was short enough so that the descendants still knew of the God of their ancestors (especially since many of these ancestors lived to see many generations of grandchildren), but the truth is more complicated than this.

We cannot speak with any absolute authority as to the average size of the post-flood families. However, we can assume that they were quite large. Genesis 10 records seven sons for Japheth, four sons for Ham, and five sons for Shem. These were the sons who lived to maturity and had children of their own. There would also be close to the same number of daughters. That would give an average of about eleven children per family that grew to maturity. Consider also the fact that the sons of Noah began their families around the age of one hundred while those generations after them began their families around the age of thirty. This, with God's command to Noah and his sons to be fruitful, multiply, and replenish the earth, would give support to the idea of the average family containing from 15 to 20 children.

For the first few generations after the flood, men were living to over 400 years of age. A man might in his old age have six, seven, or eight generations already born to him. This would tend to bring forth large numbers of people fairly rapidly. For instance, if I have 15 children who grow up and each have 15 children who grow up and have 15 children, I could easily have 3,375 great-grand-children. This would not tend to everyone knowing who great-great-grandpa was or their learning what he knows about God or anything else.

There was another event that quickly separated men from their heritage. That was the division of the people at the Tower of Babel by the confusion of their languages. This most likely occurred 101 years after the flood when there could easily have been 4,000-5,000 people trying to build a great city and resisting separation from one another. [I get the 101 years by setting the confusion of the languages at the time of the birth of Peleg. Genesis 10:25And unto Eber were born two sons: the name of one was Peleg; for in his days was the earth divided; and his brother's name was Joktan.
See All... says of the naming of Peleg: "for in his days was the earth divided." Peleg means "division." If someone is named because of something that happened "in his days," this is likely referring to the days of the pregnancy of his mother. And, it is most likely that "the earth divided" refers to the division of the peoples of the earth by the confusion of tongues (not the so-called continental drift).]

This confusion of tongues as well as the rapid growth of families would quickly separate people from their family roots and their heritage. We could assume that some basic knowledge of the true God remained while the majority of the people quickly slid toward idolatry, and this is exactly what we see in the biblical record. Romans 1:21-23 [21] Because that, when they knew God, they glorified him not as God, neither were thankful; but became vain in their imaginations, and their foolish heart was darkened. [22] Professing themselves to be wise, they became fools, [23] And changed the glory of the uncorruptible God into an image made like to corruptible man, and to birds, and fourfooted beasts, and creeping things.
See All... speaks of this time of apostasy "when they knew God" but "glorified him not as God" and thereby "became vain in their imaginations" and "changed the glory of the uncorruptible God into an image made like to corruptible man, and to birds, and fourfooted beasts, and creeping things." Therefore, it should not surprise us that Joshua declared, "Your fathers dwelt on the other side of the flood in old time, even Terah, the father of Abraham, and the father of Nachor: and they served other gods" (Joshua 24:2And Joshua said unto all the people, Thus saith the LORD God of Israel, Your fathers dwelt on the other side of the flood in old time, even Terah, the father of Abraham, and the father of Nachor: and they served other gods.
See All...). [The flood here refers to the flood plains of the Mesopotamian valley which was watered by the Euphrates and the Tigris.]

We know by this scripture that Terah and his family served other gods. How then did Abraham get to know the true God? Some knowledge of Him probably continued to exist. Abraham was a direct descendant of Eber who was still alive because of his long life. Shem is said to be "the father of all the children of Eber" (Genesis 10:21Unto Shem also, the father of all the children of Eber, the brother of Japheth the elder, even to him were children born.
See All...) and it from Eber that we get the word Hebrew (Genesis 14:13And there came one that had escaped, and told Abram the Hebrew; for he dwelt in the plain of Mamre the Amorite, brother of Eshcol, and brother of Aner: and these were confederate with Abram.
See All...). Eber would have probably known Noah and could have served as the link between Noah and Abraham. There are other evidences of a remaining knowledge of the true God. When Abraham had Sarah lie to Abimelech (Genesis 20), God warned Abimelech in a dream and he replied, "Lord, wilt thou slay also a righteous nation?" (Genesis 20:4But Abimelech had not come near her: and he said, LORD, wilt thou slay also a righteous nation?
See All...). Abraham excused his lie by telling Abimelech, "Because I thought, Surely the fear of God is not in this place; and they will slay me for my wife's sake" (Genesis 20:11And Abraham said, Because I thought, Surely the fear of God is not in this place; and they will slay me for my wife's sake.
See All...). Some of the Gentile nations still had some knowledge of God.

Therefore, Abraham grew up around idolatry but probably also heard of an ancient religion declaring there to be one and only one true God. Somehow, he came to know of that true God and received the call to leave his family and kindred and go to another place in order to establish a nation given entirely to that true God. It seems that just as the knowledge of God was disappearing from the face of the earth, God called a man to re-establish the knowledge of that true God through the nation that would come from the family of that man. What an amazing story of God's grace! And what a comfort to us today who see the apostasy of our own age. God has always had a remnant and He always will. We just need to be faithful to Him.