God tells Saul through Samuel to destroy Amalek, leaving no person or animal alive. They fight and win, but keep the best animals to sacrifice to God. God confronts Saul through Samuel - obedience is better than sacrifice and you have rebelled. Samuel wants to leave, but Saul is desperate to avoid consequences for sin and begs him to go with him to a sacrifice with the people. Samuel kills the Amalekite king himself and never sees Saul again.
God sends Samuel to Bethlehem to anoint the next king - one of Jesse's eight sons. It turns out to be the youngest, not the biggest or strongest - David. The Spirit attends the anointing and comes upon David, while Saul has an evil spirit tormenting him. Saul sends for David, who can play the harp well, and it soothes Saul.
How this is about Jesus
He was a younger brother in Israel (John 8:57).
He wasn't the biggest or strongest - obvious leader by physical appearance.
He was anointed at His baptism, to replace the current rulers of Israel and be our king.
His parents had to release Him to dangers in the world (16:20; Luke 2:34-35).
Don't judge by outward appearance.
Don't try to make up for sin with your own sacrifice. Trust Christ's sacrifice and obey.
Follow the King of Israel, Jesus Christ.
"Yahweh sees not as man sees: man looks on the outward appearance, but Yahweh looks on the heart" - 1 Samuel 16:7.