I'm a bit under the weather = I don't feel well but it isn't serious
I don't feel well
I have a cold = I'm coughing and sneezing
I have the flu
I have a sore throat = My throat hurts
I have a headache = My head hurts
I have a stomach ache
I'm sick/I'm ill = Nonspecific but more serious than "I'm not feeling well."
I hurt my back/arm/shoulder, etc.
When you call your workplace to tell them that you can't come in, this is calling in sick. For example "Bob called in sick today becuase he has the flu."
When you hear that someone is ill, you can say, "I hope you feel better soon!"