Is it correct to say "if you want to know someone' sroutine? is it better to say, "if you want to know about someone's routine?
It's much easier just not to use phrases such as "tell me" without a preposition for questions like this. There is never a need for it.
Tell me your holidays (I've never heard this) = Tell me when your holidays are.
Tell me your daily schedule = Tell me what your daily schedule is.
Tell me your routine = Tell me what your routine is.
Tell me your typical day -> Tell me about your typical day.