In my opinion, training is far more important than the contents of your first aid kit. Having a well stocked first aid kit is no substitute for knowing what to do. The more training, and the more realistic and stressful that training is, the better.

I'm currently certified as an EMT, and also as a Wilderness First Responder (WFR). While the basic skills are much the same, WFR puts more emphasis on decision making. In town, if there is any doubt about the severity of a patients injuries or illness, the EMT will immediately transport to a hospital. In a remote setting, evacuating the patient might be more dangerous than staying put. Thus WFR tends to have "Red Flags" to help one make a decision. Can this patient wait until daylight, or better weather? Or do we need to attempt a high risk evacuation ASAP? And of course EMT training generally assumes one has access to lots of equipment, while WFR stresses improvisation.

People generally don't perform as well under stress in a real situation than they do in training. Hence WFR courses tend to include a lot of scenario based training exercises, which are deliberately designed to be stressful. It is fascinating to me how a well run scenario can get ones adrenaline pumping. When you get to the point where you can perform well in a stressful scenario, you might do OK in a real deal situation.

As noted upthread, NOLS courses are good. My WFR is through WMA (Wilderness Medical Associates), who also put on a great class (usual disclaimers, satified customer etc). While there are minor differences, in general NOLS and WMA follow a fairly similar curriculum in their courses.