What was Saul searching for when he first met Samuel?

Answer: His father's lost donkeys

Kish's donkeys had wandered off. Samuel told Saul they had been found, and the next morning he anointed him ruler.

“One day the donkeys of Saul’s father Kish wandered off, and Kish said to his son Saul, “Take one of the servants and go look for the donkeys.” As for the donkeys you lost three days ago, do not worry about them, for they have been found. And upon whom is all the desire of Israel, if not upon you and all your father’s house?””— 1 Samuel 9:3, 9:20 (BSB)