We were taught in school that people did whatever it took to escape from East Germany to West Germany after World War II. How accurate is this, and are there documented cases of the opposite happening- those from West Germany escaping to East Germany?
Short answer: Yes and no. It happened in very few and special cases.
First of all we have to establish that the 1949 founded GDR (Eastern Germany) was an authoritarian regime and satellite state of the Soviet Union. Eastern Germany and the ruling Party SED built a giant surveillance apparatus and surpressed any critique at the leninist ideology and the state and severely limited free speech and human rights.
Apart from that it did a lot worse economically soon compared to the democratic and western aligned Federal Republic of Germany (western Germany) which led to a western movement by hundreds of thousands of people out of persecution and/or economic reasons by Eastern Germans. Those numbers dwindled after the GDR built the famous "Berlin Wall" (and fortified the whole border) and advised the border patrols to shoot at refugees trying to flee the regime.
Out of obvious reasons there was never a big amount of people going from the democratic west to Eastern Germany and if they did it was generally a normal case of immigration instead - the most famous example from today's view probably was Angela Merkel.
The few special cases of escape generally involve more shady people like spies and especially communist terrorists during a period called the "German Autumn". The RAF (Red Army Faction in English) and some lesser known extremistic organisations like the "Bewegung 2. Juni" went underground in the GDR when the Western German police got too close. Many of them also had financial and personal ties to the Ministry of State Security. Some of the last RAF-terrorists are still on the run today.