While not a Head of Staff, the Head Surgeon does have the responsibility of coordinating any other Surgeons that happened to show up for the shift, and ensure that they are properly handling patients. In addition, the Head Surgeon is expected to assist in matters that are out of the normal scope of surgical training (say, removing xenomorph eggs, attaching cybernetic components, etc) or when the Medbay is flooded with patients that need treatment.
The most important thing you can do as a Surgeon of any stripe is to make sure your hands and your instruments are cleaned before and after an operation. The last thing you need is a patient complaining of sepsis or necrosis, so try to avoid that by properly washing your hands with soap, and putting your instruments into the sterilizer to clean between patients. In emergency situations, you can sterilize instruments with chemical treatments, but bear in mind that the chemical will remain on the instrument - this can work to your advantage in some cases, less so in others.