*sag-

The core idea of this root was:

tracking, seeking

This is well preserved in the word to seek and the German suchen, but a second branch has shifted toward the idea of debating, arguing, quarreling – maybe the theme was “seeking truth” here.
That’s where the word sake and the German die Sache come from, which originally was matter debated.

The word to search does NOT belong to this family and instead comes from the Latin branch around circle. So searching originally was about going around in circles.