Everything is attending or resident dependent so if the attending or resident says: "Do it like this!" Never argue with them, just do it that way because there is more than one way to practice and it is better to learn multiple ways than just one way. Who to go to is totally attending depedent. Sometimes I have students report directly to me, sometimes I have them go to the residents and just listen in while charting on something else. Generally speaking though, I think most attendings and residents would want more pertinent information faster to get the ball rolling on the labs/imaging than every single detail explained. Also, while for you going back and asking more isn't always a great option, for the resident who has 5-6 other patients and the attending who has 10-15, it's a blessing. EM is all about pertinent information. Obviously this is a skill that is developed over years. Some more tricks:
1) Look at vitals before you go into a room. This will allow you to triage the patient mentally and ask focused pertinent questions. If the patient is being triaged with you in the room assist in obtaining them. This will expedite the process and also establish excellent rapport with nurses who are usually trying to do multiple things at once (careful though, some nurses bite).
2) For the patients with complicated PMHs, pull up a prior H&P or discharge summary. This is basically a cheat sheet. Obviously in a patient who twisted their ankle, this is overkill. In the demented septic patient who can't answer any of your questions, this is a huge time saver. Do this if a patient is waiting for a room or right before you go in to see them.
3) Don't let patients ramble. You don't have to be rude but you may have to redirect. This may involve asking the same question multiple times.
4) Tell your attending/resident up front: "Hey I got the basics on the patient in 9 so we can put in orders, I will go back and ask any additional questions if you think I missed anything." Most normal EM docs will appreciate this approach.
5) Think of the things that could kill someone with this chief complaint, then work backwards.
6) Focus your RoS and exam to the pertinent positives and negatives.
I would also echo
@Daiphon in saying that sometimes missing something is a good thing. You learn from it because you're a hard worker and you don't like being wrong and it shows that you are willing to admit that and show a desire to improve. Just never lie. Ever. We know that stories change from patients from one minute to the next. It happened to us as students. Don't worry about that. Just don't lie. Ever.