As a native English speaker I have no problem with "I am going east." "I am going to the east." is acceptable, but not mandatory.
Also to say "to the east/south/north, etc. sometimes implies the region one is going to rather than merely the direction. In fact, one might say "I am going to the East" if going to China, even if the plane is flying west across the Pacific.
Doğuis related to the verb doğmak = to be born, to emerge, to rise – cf doğum = birth, doğa = nature. Doğu is the point of the horizon where the sun is born, where it rises.
Doğrumeans not only right as in correct and true but also right as in straight and righteous. As a noun it means line. It comes from the Old Turkish root toġ– and is related to such words as the noun direk = column, pillar and the adjective dik = upright, steep.