먼저 -" come first" is the same as Chinese word 先 ( 선) as in a phrase 선생님 - 先生님 which literally means that the person of respect who was born first.
The most commonly used phrase in daily life is 먼저 가겠습니다. - I am going to leave first. You say it in formal situations when you are leaving before the others. You can hear it a lot in korean dramas.
Another common phrase is 먼저 하세요 - Please do first/ after you