Breathing, bleeding, beating and broken, in that order, I think. (It’s been a long time since I studied First Aid, but any First Aid manual should give you the order of things to check for.)
If he’s not breathing, he has a minute or two to start or he’ll begin to suffer irreversible brain damage and then death, in that order and in close succession.
If he’s breathing, then he can still bleed to death (and depending on the flow, such as from a severed main artery, it can be a race with breathing). Since failure to breathe will always kill soon, and heavy bleeding “may” cause death almost as quickly, an experienced First Aider will always check breathing first, and then blood loss.
Next, is his heart beating? If he’s breathing and the blood loss is minimal, then his heart has a bit of leeway (almost as critical as breathing, but not quite). If his heart isn’t beating, then I think CPR techniques are changing now, and you wouldn’t do chest compressions and artificial respiration, but only the chest compressions, since that will do some “respiration” naturally on its own. (I think you’d want to know when you checked for breathing if his heart was still beating, so you might go right into the chest compressions if that were required.)
If he has broken bones, especially spinal or cervical (neck) bones or a possibility of that, say after a fall, then you may be advised to attempt to keep him comfortable (warm, dry, as safe as possible, feet elevated slightly to allow blood to pool in the trunk area), but to not move him.
A first aid manual should really tell you what you want to know about “technique”. The other thing the 911 operator will check is your address, even if their caller ID “shows” where you are, they will want to confirm that that is the address to respond to, and can someone be available at the street to flag the ambulance & police responders.
If there’s any possibility that this was NOT an accidental shooting, or if the victim appears or has appeared to be drunk, drugged, despondent, etc., then they’ll want to give any information available to the police so that they can secure the area. Does he have any more weapons? Are you with him right now? Can he respond?
The operator will request that you stay on the line until help arrives.