-Turn (optional pronoun) on=Sexually arouse (optional pronoun). For example: "Tom really turn me on when he plays romantic music." -Turn (obligatory pronoun) on to (obligatory pronoun/noun)=Get (obligatory pronoun) interested in (obligatory pronoun/noun). For example: "I really like the band Death Grips. Tom turned me on their music." The meaning in your sentence is the second meaning. It has nothing to do with the verb "to turn".

If I'm not mistaken, it means that the writer started reading this Spanish poet upon his/her sister's advice.

It's not actually a phrasal verb. "To turn" here just means "to direct one's attention". "To" indicates the where 'my' attention was directed towards, and "on" just adda extra detail to "to".