disease is something infectious.
"sick" and "ill" depend on which version of English you're using.
In British English if you say "I'm sick" you usually mean stomach problems, hence "morning sickness". "morning sickness" is also used in AmEng.
In AmEng if you say "I'm sick" you mean "I'm ill", but if you say "I'm ill" you mean that you are really sick/ill with something serious. So, a common cold would be just a sickness, whereas a pneumonia would graduate to an illness. And a flu might be somewhere in between.
You can also say "I'm sick" in AmEng to mean "I'm nauseus" - same as in British English. So if you say in AmEng "I'm gonna be sick" you mean "I'm gonna puke". But you can also say "I'm gonna be sick" and mean "I'm going to be sick/ill" if you think you're coming down with a cold or something.