"vorsorgen" is "to take precautions":
Ich sorge für mein Alter vor.
"umsorgen" is to look after someone, but to do it with special fervor:
Wenn ich als Kind krank war, hat mich meine Mutter liebevoll umsorgt.
"kümmern", on the other hand, may be used if you just take care of something:
Ich habe mich um meine Steuer gekümmert (I took care of my taxes)
Because of the edit:
"versorgen" means "to supply". When used to speak about care, I'd say it has a more formal connotation: a mother "umsorgt" or "kümmert", but a nurse "versorgt".