"Please read me the story" and "Please read the story to me" are both fine, in fact.
But I would never say "Please read to me the story." While it IS possible to say "Please read to me" (or "Please sing to me"), this isn't the same as saying "Please read to me the story."
The difference is that "Please read to me the story" contains both a direct object ("the story") and an indirect object ("me"). When we have sentences with both direct and indirect objects, they follow either of these constructions:
SUBJECT + VERB + DIRECT OBJECT + TO + INDIRECT OBJECT
John + gave + the book + to + me
OR
SUBJECT + VERB + INDIRECT OBJECT + DIRECT OBJECT
John + gave + me + the book
The same is true for "Please read the story to me" or "Please read me the story." You cannot say "Please read to me the story" because "to" is not used if the indirect object comes before the direct object.