It depends what situation you're talking about:
When you are a resident, you will call your attendings "Dr. _____" unless they insist otherwise. You will be called by your first name, or if in a surgical field, by a rude offensive nickname.
With your fellow residents, you will likely go by first names or nicknames. I recommend initially calling your senior residents "Dr. ______" but that won't likely last, except maybe with the Chiefs.
When you introduce yourself to patients, it is crucial that you introduce yourself as "Doctor". Patients don't want to be your friend, they want to have confidence in you. Introducing yourself with you title will make your role clear, and inspire trust from your patients. Nurses will introduce themselves by their first name, and will likely also wear scrubs and white coats. You don't want to be confused with a nurse. I highly recommend you call your patients "Mr. ______" or "Ms. _______" to be proper.
Personally, I refuse to go by first names with nursing and allied health. On the other hand, I recognize that introducing yourself as "Doctor" is considered somewhat pretentious. I introduce myself by my last name only. What you do is your business, but I tend not to like to get too friendly with them.
And remember, only a douche registers at hotels and restaurants as "Doctor".
.. it sounds similar to my last name so thats where they got it from, and even i though it was amusingly creatively sick.
