There're several ways of saying "I'm bored", but 심심하다 is most common. Actually, I'm a bit amazed that this word hasn't been mentioned. 심심하다 implies that you're bored because you have nothing special to do. It's one of my favorite words. 지루하다 is less common probably because it already sounds boring itself. 따분하다 mean "I'm bored" or "It's boring". 지겹다, 싫증나다 is closer to "I've had enough of it" or "I'm sick and tired of it".

I'm bored 1. 지루해(You can tell a familiar friend.), 지루해요, /지루합니다(The most polite expression) 2. 지겨워(You can tell a familiar friend.), 지겨워요, /지겹습니다.(The most polite expression) Many korean, Almost say except subject. You don't need to say like 나는 지겨워, 나는 지루해 ... blar, blar'

You can say "지루해(지루해요)" , "따분해(따분해요)" , "지겨워(지겨워요)" or "싫증나(싫증나요)" .