Textbook vs Real
Textbook안녕히 가세요an-nyeong-hi ga-se-yo
Real먼저 갈게~meon-jeo gal-ge~
Why Koreans say it this way
안녕히 가세요 is for strangers/elders. With friends: 먼저 갈게 (I'll go first), 나 간다~ (I'm going~), or just 빠이~ (bye~, from English). The ~ adds softness.
Situation: social