病気です。 means I have an illness. Not illness such as catching a cold. A bit serious one which you need to see a doctor with.
具合が悪いです。 means I'm in the bad condition as Jasmine says. We say that especially when having a headache/stomachache/diarrhea, feeling nauseous, or when you can't get away from your bed due to illness.
気分が悪いです。is used in 2 ways: when you feel nauseous and physically sick, and when something bad happens to you.
Imagine when you ride the most terrible roller coaster several times. Your head would swim and you feel nauseous, wouldn't you? Then you would say 気分が悪いです。.
And imagine when you see a crime scene with lots of blood, then you would feel worse or nauseous and say 気分が悪いです。. So you can say it when you see something gross or ghosts.
One more example! Imagine when you get chewed out by your boss or teacher unreasoningly, then you would not get the reason,feel bad and complain 気分が悪いです。to your friend or family.
We never say 病気です。and 具合が悪いです。when something bad happens to us.
-Other ways to express depression or sadness-
辛い(つらい) Very useful expression when you're sad or having loads of work to do, a part of your body really hurts and so on.
病んでいる(やんでいる) A kind of slang, but useful and 2 ways to use. One is when you have a serious illness especially depression. Another is when you are depressed and very worried. It comes from a verb 病む(やむ)。
苦しい(くるしい)This is also useful and similar to 辛い.
がっかりする It means to be dissapointed and the same with 落ち込んでいる。
That's all we often say, and the words Jasmine says are also quite common.