In Australian English, kindly implies a polite instruction (which is expected to be obeyed) rather than a polite request (which may be ignored or rejected). Accordingly, kindly is usually used by someone in a position of authority. The formality also implies that the person may be irritated.
If a coach or manager says to a football player:
'Kindly practise every day,' he means that he expects the player to do so (and implies perhaps that the player has not done so in the past).
'Please practise every day' means that he wishes the player to do so.
Teacher to student: Either is correct but kindly implies exasperation or irritation.
Student to teacher: The use of kindly could be considered rude as it places student over teacher.
Between siblings: Please would be used in an Australian cultural context.
Because of its formality, (this dictionary lists it as old-fashioned); it would rarely be used among peers.
I expect British English would be similar but I cannot be certain.