The expression "fall ill" implies a sudden onset of sickness, which can range from a mild cold to a severe illness. Broadly, some synonyms for "fall ill" are "get sick", "fall sick" and "take ill". However, when describing specific types of illness, different synonyms may be more appropriate. For instance, "become feverish" or "develop a cough" could be used when talking about specific symptoms. "Contract a disease" or "come down with an infection" might be used to suggest a more severe illness. On the other hand, to describe a milder form of sickness, "feel unwell" or "get under the weather" could be good alternatives.