This construction indicates that the second action has a dual purpose. Taking advantage of the situation, you do one thing while doing something else at the same time. 

• 선생님도 뵐 겸 학교에 갔다. 
  => I went to school and at the same time visited my teacher.
  
• 친구도 사귈 겸 한국 학생회에 나갔다. 
  => I went to attend the Korean students' meeting and also to make friends (there).
  
• 점심도 먹을 겸 집에 갔다.
  => I went home and also had lunch. 

-더니 is not used with I or we. This restriction does not apply to -었/았더니.

(a)

• 수잔이 한국에 가더니 소식이 없네. 
  => Since Susan went to Korea, there has been no news. 
  
• 인류학을 공부하더니 지금은 사회학을 공부한다.
  => She was studying anthropology, and now he has changed to sociology.

• 아침 일찍 학교에 갔더니 교실이 텅 비어 있었다.
  => Because I went to school early in the morning, the classroom was completely empty.
  
• 전화를 했더니 친구는 벌써 나가고 없었다. 
  => I called a friend, but he was already out.
  
• 밥을 많이 먹었더니 배가 아프다. 
  => I ate too much and have a stomachache. 
  
• 우리가 공부를 열심히 했더니 시험을 잘 봤어요. 
  => Because we studied hard, we did well on the test.

Note that the subjects of the -었/았더니 connective are usually not the same. For example, 톰은 책방에 갔더니 책을 샀다 is not correct, but 톰은 책방에 가더니 책을 샀다 or 톰이 책방에 갔더니 민수가 와 있었다 is fine.

This expression indicates the speaker's conjecture or reasoning.

• 바빠서 그런지 죤은 무술에 관심 이 없다. 
  => Maybe because he is busy, John is not interested in the martial arts.
  
• 잠을 못 자서 그런지 머리가 아프다. 
  => It could be that because I didn't sleep, I have a headache.
  
• 학생이 많이 늘었다. 그래서 그런지 갑자기 교실이 모자란다.
  => The number of students has increased. Maybe that's why there aren't enough classrooms. 

This construction indicates that the speaker recognizes the fact.

• 비행기가 빠르기는 하지만 비싸다. 
  => It is true that airplanes are fast, but they are expensive.
  
• 데이비드가 요리를 맛있게 하기는 하지만 시간이 없어서 잘 하지 않는다. 
  => It is true that David is a good cook, but he doesn't cook because he doesn't have time.

(b)

• 전통 무술이기는 하지만 잘 몰라요. 
  => It is a traditional martial art, but I don't know it well.
  
• 좋은 선물이기는 하지만 저는 못 받겠어요. 
  => Although it is a nice gift, I cannot accept it.

There is no equivalent expression in English for 뭐. It is a colloquial sentence-ending expression meaning something like "what else?" in 나 학교에 갔지 뭐 "I went to school, what else?" or "I went to school, you know." It is used only in intimate and informal conversation and follows the -지(요) or -나(요) sentence ending.

• 일종의 스포츠로 하는 거지 뭐. 
  => They are doing it as a kind of sport, nothing more. 
  
• 오늘 안 오려나 보지 뭐. 
  => I guess he's not coming today, you know. 
  
• 학교에나 가지요 뭐. 
  => I'll just go to school, nowhere else. 
  
• 제가 어디 가나요 뭐. 
  => I don't go anywhere, you know. 

This intimate or polite sentence ending says something like "you know," as in "I saw the movie, you know."

• 무술은 건강에 좋거든. 
  => You know, martial arts are good for your health. 
  
• 내일은 휴일이거든. 
  => Tomorrow is a holiday, you know. 
  
• 꼬마 동생 정엽이가 숙제를 찢었거든요.
  => My baby sister Jung-yup ripped my homework, you know.