Some conditions have different response. For example when someone dies we say condiliansa, when we welcome someone we say bene venuti, But when someone get a disease what do we say to him?
I would not say "tanti auguri" to someone who is ill! That sounds weird.
I would say: "Guarisci presto!" or "Rimettiti presto!" (2nd person singular, that is, for people you address with "tu") / "Guarisca presto" or "Si rimetta presto" (3rd person singular, that is, for people you address with "lei"/"signora"/"signore).
Rimettersi means "get better".
If you need to be very formal, like if you are writing a card to bring to someone who is in hospital or very ill, you can write: "Auguri di pronta guarigione". This is ok for people you address with "tu" and "lei".