“Now the LORD said to Abram, “Go forth from your country, and from your relatives, and from your father’s house, to the land which I will show you.” Gen. 12:1
Abraham left his country (which was either Ur of the Chaldeans, or Haran, depending whom you ask), but he did not leave his relatives. Probably facing a difficult decision, Abraham decided to bring his nephew Lot to Canaan. Lot was presumably younger, and was fatherless since his father Haran died. And Terah may or may not have passed away at this point. Abraham probably thought it best to adopt Lot rather than leave him to fend for himself, even if against God’s command. Besides, Abraham had a group of people already going with him.
This decision was probably not without regret. Lot caused some troubles for Abraham. For one, he took the best land, the entire valley of the Jordan (Gen. 13:11); that after infighting between his herdsmen and Abraham’s herdsmen (Gen. 13:7). Then Abraham and his men had to act as an impromptu army to save Lot, who had been taken as prisoner in the war of the kings (Gen 13:12-14). Lastly, Abraham took up argument against God Almighty for the sake of the cities of Sodom and Gomorrah, where Lot was living at the time – an argument which proved useful, at least for Lot, who was ultimately saved by angels and spared the destruction.
As I mentioned, it is unclear if Abraham’s father Terah died before Abraham left or not. Rambam notes that it is common in the Torah to record the death of a father before proceeding with the narrative of the son (as was the case with Noah – he was still alive at the tower of Babel though his death was already recorded.) Rashi notes that Abraham left Haran more than 60 years prior to Terah’s death. So it is unclear if Terah was truly in old age, and or bad health, if and when Abraham left. On one hand it would be considered disgraceful to leave an aging father before his death; on the other hand, the command of God to go forth and leave his relatives behind, would have superseded the traditional expectation here.