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traveled to the Judean town of Bethlehem
This could be translated as "went up to the town of Bethlehem in Judea." Bethlehem was higher in elevation than Nazareth.
city of David
Bethlehem was called a city because of its importance, not its size. King David was born there and there was a prophecy that the Messiah would be born there. "City of David" could be translated as "city of King David."
to register
This could be translated as "to have his name written in the register" or "to be included in the count."
along with Mary
Mary traveled with Joseph from Nazareth. It is likely that women were also taxed, so Mary would have needed to travel and be registered as well.
who was engaged to him
This could be translated as "his fiancée" or "who was promised to him." An engaged couple was considered legally married, but there would not have been physical intimacy between them.