"According to" is used to refer to something, whether a belief, idea, or account of events, as it was expressed by somebody else. E.g. "According to our teacher, that is the correct answer." In this case, the answer may or may not have actually been correct, but in the teacher's eyes, or according to the teacher, it was.
"Regarding" is simply used to bring your audience's attention to something. It literally means "regarding" in that sense. E.g. "Regarding last night's homework, I was made aware that a few of you did not complete your assignments." E.g. "She had no idea what to do regarding the predicament."