Hi FastFred,
After you light up the yellow blobs on the ship by clicking the top (bollard) mooring button, you left-click on the ones to which lines are attached. That drops the line.
Lines can be hard to see; a planned change in the next patch will identify the blobs that have lines attached. For now: If you click a blob that has no line, the blobs on shore will light. Click that bollard button to re-light the ones on the ship, and try another blob.
If you drop all the lines, the ship should not sink as you pull away.
As the other Fred said, the manual gives a quick explanation on how to operate most of the controls.
Regards,
Marty