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Sentences you wrote in Korean
춤을 배우다.
한국어를 공부하다.
다른 사람에 대해 알다/알아보다.
배우다 is to learn
공부하다 is to study
알다 is to know
In English, learn has broad meaning, but in Korean it is more specified.
배우다 - when you learn from someone else(generally teacher)
공부하다 - when you study by yourself(when you are learning from a teacher, it should be 한국어를 배워요)
알다 - if you mean learn about some historical person it would be either 배우다 or 공부하다, but if not, then you're knowing/ trying to know someone else and in that case we use 알다/알아 보다 which means 'to know'.
Do you get it now?
