to seek: to go after something with effort, whether searching for it, striving to get it, or insisting on having it.
At its heart this is one simple idea: to direct your effort toward something you do not yet have. What changes from verse to verse is the object and the intensity, not the core action. You can seek a lost coin, seek to do something, seek to put a person to death, seek a sign as proof, or seek God and his kingdom. In every case someone is reaching for something not yet in hand.
Because the word covers such a wide reach, English Bibles spread it across many verbs: look for, try, want, demand, require, strive after. That range is not a set of separate words but the same verb pointed at different goals. The shades below are sorted by what is being sought, since that is what gives the word its color in a given sentence.
The Gospels lean on it heavily for two opposite pursuits. On one side it names the hostile search for Jesus: the authorities repeatedly seek to seize or kill him. On the other side it names the pursuit that matters most, the call to seek God, to seek the kingdom, to seek and you will find. The same plain verb carries both the plot against Jesus and the invitation to come to God.