to seek something out with care and effort, going after it deliberately until it is found, answered, or settled
The picture under every use of this word is a person turning toward something and pursuing it on purpose. It is never a vague wish; it is the act of going after a thing, tracking it down, asking after it, looking into it, until you get an answer or a result. Because the same effort can be aimed at very different targets, the word fans out into several settled uses.
The oldest and most common target is God. To seek the LORD is to go to him for direction, to consult him through a prophet or by some appointed means, and more broadly to make him the object of one's loyalty and worship. The opposite act, going to dead spirits or other gods for the same kind of answer, is described with the very same word, which is part of why the Bible warns against it so sharply.
The other uses keep the same core but aim it elsewhere: searching out a lost thing or a person, looking carefully into a matter to find out the truth, holding someone responsible and demanding an accounting, and tending to the wellbeing of those in one's care. What ties them together is the deliberate going-after, not the particular object.