The red line (that can fade-out, or come closer) is an enemy warship, picked up by the way you set your realism settings, or newer sub radar is picking it up. To get him mad just go flank-speed and chase the line!
The small/single ship icons are usually close contacts (supposed 'spottings') that relay this spotting-info to you. If it's a red ship icon, you can plot a course and try to catch up with it, submerge before seen, and sink the ship.
Be sure to complete the 'Naval Academy' missions and movie tutorials for each. They really help understand the campaign/patrol parts of the game. Completion of all Academy missions also gives bonus renown points for each newly created campaign.