How do we wish someone to get better?
One common phrase is this one, where X is the somebody in your question:


  祝X早日康復/祝X早日康复。
  Zhù X zǎo rì kāng fù


For example, 祝你早日康復/祝你早日康复, which can be translated into "wish/hope you'll recover soon." or "wish/hope you'll make a speedy recovery."

Vocabulary:


  
  祝 (zhù) = Wish
  你 (nǐ) = You
  早日 (zǎo rì) = in the near future, soon
  康復/康复 (kāng fù) = recover from illness
  


This phrase can be used in most situations (mostly formal, informal and situations 1, 2 &amp; 3 in your questions).

For "light" sickness, say a cold or a headache, people often say:


  
  多休息/多休息 = Take more rest.
  Duō xiū xí
  多保重/多保重 = Please take care.
  Duō bǎo zhòng
  小心別又著涼了/小心别又着凉了 = Please don't get cold again.
  Xiǎo xīn bié yòu zháo liáng le
  


You'd often hear parents or elderly relatives say Example 3 to their kids or young people in the family when they have a cold.