Anesthesiology, like radiology, is one of those specialties where you can't get a lot of responsibility as a medical student. It's different enough from your bread-and-butter internal medicine, for example, that everyone knows students come in not knowing a thing about anesthesia. So the expectations are:
1) You don't know anything, and you don't necessarily need to start out knowing anything to impress.
2) You don't have to scut to impress.
In my experience, the best way to get the most out of an anesthesia rotation is:
1) When an attending comes in to give a break to the resident or CRNA, stay around and ask questions about the case. (It helps to carry around a pocket manual of some sort so you can read the outline about how to manage that case -- I started out using NMS in my first rotation, which is tiny and cheap but very sufficient for a first-time, and eventually moved on to the Massachusetts General Hospital handbook.) Take your break after the resident returns.
2) Stay for the entire case if it's a short case. If it's a long case, do as the person in the room suggests to you (some might say leave, go to another room to get some procedures, and return for the end of the case).
3) Think of teaching topics to discuss and ask questions. Ask lots of questions, questions are appreciated and it's not how much you know but how your questions show you've thought about the case that really matter.
4) Read something from time to time and ask questions about it (shows you're reading). The first thing I read was the UpToDate article on how pulse oximetry works -- nothing thorough and the article wasn't even written by an anesthesiologist -- but again it's not what you know but what you're curious about that matters.
5) Look for ways to have more patient care responsibility even though you'll have little in anesthesia. You do learn a lot from these experiences, especially if you push for someone to assign them to you.
--> Meet the patient in the pre-op area and go through the anesthesia pre-op checklist, asking questions like when was your last meal, etc. DO THE PHYSICAL EXAM (bring a stethoscope, do a heart-lung exam, and do an airway exam).
--> Learn to put EKG leads on (very useful, whatever specialty you pick).
--> Offer to round on post-op and epidural patients (you can learn a lot, especially if you do them on your own).
--> READ THE OLD CHART and look for old anesthesia records.
--> If your hospital has paper records, ask to be taught how to chart (you'll learn a lot).
--> Choose one or two specific tasks to be responsible for during a case (such as checking the eyes/nose, monitoring the urine output, charting, etc) and do it for the whole case (again, you'll learn a lot and it'll help you learn to chart).
--> BE AROUND FOR THE END OF THE CASE if possible -- e.g. if you intubated, extubate!
--> Ask to be the hands for "non-patient procedure" things like drawing up drugs, hanging fluid, hanging blood, etc. (You really, really don't know how to do these things when you start out as a medical student. Do them, and it keeps you from getting bored during the case.)
so I hope that's a decent start...
