Quoted from
http://www.talktomeinkorean.com/lessons/l6l14/
겸
When you want to say “breakfast/lunch” or “singer/actor”, you use the word 겸.
Examples
1. 아침 겸 점심
= breakfast/lunch, brunch
2. 가수 겸 배우
= singer and actor, singer/actor
3. 화가 겸 작곡가
= painter/composer, painter and composer
-(으)ㄹ 겸
When you want to talk about an action that has two or more purposes, you can list them using -(으)ㄹ 겸. When you just mention one purpose using -(으)ㄹ 겸 in a sentence, the other purpose has to be understood from the context. And since -(으)ㄹ 겸 usually has the nuance of “both at the same time”, you often use the particle -도 (meaning “also”) after the noun that comes before “Verb + -(으)ㄹ 겸”.
Examples
1. 바람 좀 쐴 겸 밖에 나왔어요.
= (There’s another reason/purpose, too, but) I came outside to get some fresh air.
2. 영어 공부도 할 겸, 영어로 된 소설을 읽고 있어요.
= (I’m doing it because it’s fun, too, but) I’m reading a novel written in English to study English as well.
3. 친구 생일 선물도 살 겸, 제 옷도 볼 겸, 백화점에 갈 거예요.
= I’m going to go to the department store to buy a present for my friend’s birthday and, at the same time, look at some clothes for myself.
4. 산책도 할 겸, 사진도 찍을 겸, 남산에 갔어요.
= I went to Namsan to take a walk and also to take some photos.