Right way means do things "the way it should be done". When your parents bring you the right way, it means that your parents have taught you all the correct values and morals to make you become a responsible person in this society and amongst your peers.

In Aguilera's song, since I don't listen to her songs and haven't read the entire lyrics for that song, so I'm not quite sure what she meant there. But we do have an expression in English that goes like this "You gotta rub that guy in the right way to make him happy". Meaning that you gotta know the person's "soft spot" and get him with his soft spot to make him a much accessible or friendly person.

This is as best as I can explain. Hopefully it will be helful to you.