For most fights, you’re looking at bruises and minor cuts.
With bruises you want to wrap some ice in a towel and apply it to the injury, and let it heal on its own. Strictly speaking, bruises are minor, sub-dermal hemorrhages. There are rare cases where someone loses enough blood from bruising to die, but this is usually accompanied with massive amounts of trauma.
Also, it’s worth noting, it usually takes about five minutes for a bruise to start to show, if someone is killed within that timeframe, the bruise will not develop.
Minor cuts can be treated with peroxide or alcohol (usually rubbing alcohol, but anything over about 40 – 60 proof s work) to disinfect the wound, and then bandaged. Applying a petroleum jelly like Vaseline or Neosporin can help keep the wound clean. Strictly speaking, it doesn’t actually do anything, the stuff’s completely biologically inert (and as I recall, technically edible) but it will prevent new bacteria from getting into the injury.
More serious cuts, like knife wounds, can require surgery, as far as I know, this is just another round of disinfect the wound, and then stitched closed. If there’s internal damage, that will need to be dealt with, based on the injury sustained.
Gunshot wounds are an entirely different and very unpredictable animal. Again, I’m going to over simply here, so, apologies in advance to any doctors out there. The primary threat from any gunshot wound is bleeding to death. If the gunshot damaged an artery, then first aid involves compressing the artery to staunch the blood loss, this is a very basic, and limited stopgap. Otherwise “first aid” is getting the victim to a doctor (or veterinarian). The basic surgical techniques to deal with gunshot wounds is to repair whatever damage the doctor can, and closing up the wound. But, this can be a lot more difficult than it sounds.
I’m going to split gunshot wounds into three general categories, these aren’t official classifications, and shouldn’t be held up as holy writ; this is just an attempt to get everything out in a readable fashion: blowthroughs, ricochets, and fragmentation.
Blowthroughs, are the “best”, and most common kind of gunshot wound. These are gunshots that enter the victim, pass through them, and leave for parts unknown. If it’s a headshot, the victim is probably already dead, though, there are a few medical cases where people survived a shot to the head. Blowthroughs to the torso usually mean punctured internal organs, regardless what organ was hit; the injury will require major surgery to deal with. If it’s a hit to the limbs, and it missed the arteries, the wound will need to be sewn up, and cared for. The limb can’t be used for a couple weeks. If the bullet nicked or severed the artery, the surgeon will need to repair it, assuming they get the chance. A damaged artery can result in the victim bleeding to death in minutes. This, by the way, is what the whole “apply pressure here” cliché is referring to; first aid for an arterial hit is to apply pressure and staunch the flow of blood, so the victim can live long enough to reach a doctor.
Ricochets are cases where the bullet connects with a bone and reflects off in a new direction. This is highly dependent on the specific physics involved, but the result can be very messy. The best case scenario, a bullet will ricochet off a bone, and have a clean exit wound. This is slightly more problematic than it sounds; usually, you track the path of a bullet by checking the entrance and exit wounds, with ricochets, it can be very difficult to identify which internal organs have been injured. Worse, it’s not unheard of for bullets to start bouncing around inside the rib cage, tearing someone’s internal organs to pieces. It’s rare, but can result in irreparable internal damage.
Fragmentation refers to where the bullet breaks apart into multiple pieces. Usually this is associated with fragmentation rounds, also called dumdums, but a bullet that impacts a hard object, either bone, or something outside the body, can shatter; sending shrapnel into the victim. The shrapnel is slightly more prone to further ricochets and lodging in the body. As with ricochets, this can result in massive internal injuries that will require extensive, rapid surgery to survive.
One last note: In modern contexts, it is fairly common to get hit by multiple bullets in rapid succession, called a multiple gunshot wound (or MGW in police and paramedic reports), multiple bullets effectively multiply the damage. Because the victim is bleeding out faster, a doctor won’t have time to treat the victim before they expire, assuming they’re able to hold on long enough to get to a doctor.
If you’re writing about first aid for hand to hand, I’d actually recommend you look into first aid techniques first hand. It’s a useful skill to have, and it should be fairly easy to find a reputable group that’s teaching the basics.
You can learn far more than you want to about gunshot wounds in most forensic texts. I’m not sure where you’d find more detail on the specifics of surgery. Generally speaking finding information on specific surgical procedures, which are also accessible to a layman, is tricky.