What is Judaism's view on illnesses? Is the point to punish the person who gets sick, as found with Chizkiyahu, who didn't listen to God about getting married (and I think Yeshayahu rebuked him and said - bhadai kivshei rachmana, lama lach?) and therefore was punished? Or are there other reasons why God would make someone sick? If anyone has sources on this, that would be much appreciated.
here's a source from the shaar bechina ch.5 of chovos halevavos regarding why children get sick. according to the marpe lenefesh commentary it also applies to adults too.
Later on he is subjected to illnesses and meets with painful incidents so that he recognizes the world, and that its nature is not concealed from him. Thus he is put on his guard against trusting in this world thereby permitting his lusts to rule over him, in which case he would become like the animals that neither think nor understand; as it is written "Be ye not as the horse or as the mule which have no understanding" (Tehilim 32:9).
(Tov Halevanon: "illnesses" - such as chicken pox and measles. "painful incidents" - many weaknesses come in the boyhood years. PL - accidents such as stepping on a metal nail. "Be ye not as the horse or as the mule" - they need to be leashed and muzzled, so too man. The painful incidents humble his lusts. "so that he recognizes the world" - how a person's situation can change swiftly from contentment to pain so that he realizes not to trust in it and its tranquility - rather to always be afraid and to seek refuge in G-d's shadow.
Marpe Lenefesh: If a human being had only constant good in this world, he would forget and not recall the matters of his final end, and he would trust (hope) in this world and follow the musings of his heart and his lusts for all of his days. Therefore, it was among the divine plan to send him sometimes bad illnesses, EVEN during his youth in order that he recognize and know that there is no complete good in this world. And even if he is in a very good situation, the bad illnesses can come and ruin his joy, so that he won't trust in this world.)
emphasis on the word "even" is mine
while there may be other reasons for specific cases, I think this is the most common one.