In which direction does a submarine travel to escape attack?

It depends entirely on the tactical situation, e.g., the type of threat (torpedo, helo, sonobuoy / ship tracking, etc.) as to which direction to escape. With torpedoes, the tactic has long been to keep them on the edge of the sonar baffles (astern), but with modern homing torpedoes, once they get a lock on you, you're pretty much done for.

In general though, most escapes are away from the attacking vessel and to deeper water to use the ocean environment as acoustic cover. Remember that with today's boats, their speed and maneuverability underwater can match most surface ships that are attacking, and can escape an area pretty quickly.

The biggest threat is aircraft (helos and aircraft with sonobuoys), but a good sonarman can hear aircraft as well if conditions are right.