Well, I don't think my response is the absolute answer to your question, but these are some bullet points off the top of my head that anyone carrying should have ingrained in them:
1) A very good knowledge of when it's appropriate to draw your weapon and, more importantly, when it's not.
2) Gun handling basics should be second nature. Meaning how you hold the gun, where you place your trigger finger, etc.
3) A thorough knowledge of your particular gun. It's safety mechanisms, firing and loading mechanisms, how to clear jams safely, etc. These should be second nature.
4) How to safely carry the gun on your person. Too many idiots just tuck it into their waistband and call it good. Or leave a round in the chamber at all times. The guy who shot his kid recently is a sad example of why that's a bad idea.
5) How to draw the gun, and thorough practice doing so.
6) This might seem obvious, but knowing how to shoot. Not just being a good shot, but knowing under what scenarios actually pulling the trigger is justified.
The list can go on and on, but I think these are the basics.
And I think the "vigilante" idea is a misperception, or should be. Nobody should be carrying a gun with the intent of being a vigilante. It's a tool of self-defense, and I would argue it shouldn't be used in any other scenario by the average citizen.