"I do not care what you think."
Translation:Mir ist egal, was Sie glauben.
it might help as a sort of bridge to your understanding if you think of "mir ist egal" along the lines of "it's the same to me" (i.e. to you're indifferent to whatever is at stake) which brings you closer to "I don't care". In which case, "Mir ist nicht egal" would then amount to something like, "it's not the same to me" implying "i do care [what you think]."
The verb ‘sorgen’ means “care” in the sense of “taking care of” someone, as in ‘Ich sorge für Dich’ = “I'm taking care of you”. The noun ‘Sorge’ means “care” in the sense of “worry”, as in ‘Ich mache mir keine Sorgen um deine Gedanken’ = “I'm not worrying about your thoughts.”. But there's no usage of ‘sorgen’ corresponding the the sense of “give a damn” in the sentence “I do not care what you think.”