What’s in our first aid kit (which is different from our medicine cabinet):
A selection of Band-Aids in different sizes.
A few of those big gauze pads, in different sizes.
First-aid tape, to hold the pads on.
Scissors, to cut the tape.
Triple antibiotic ointment.
Tweezers.
And that’s it. I already have needles for splinters elsewhere in the house, and rubbing alcohol, too, if I need to disinfect the tweezers and needle.
You wash out a wound with plain soap (mild dish soap or antibacterial liquid hand soap works well) and warm water.
If somebody’s got an owie that a gauze pad and plenty of tape can’t deal with, then you’re talking the hospital ER and stitches anyway.
And actually, the antibiotic ointment isn’t mandatory. If you feel you must use it, let the owie dry out and form a scab before you use the ointment; in my experience, if you put the ointment on while the wound is still raw, it never does form a good scab, but stays moist and oozing. Let it dry out a bit first. Clean it good with soap and water, dry off around it with something reasonably clean and mostly sterile like a paper towel (i.e. not a washcloth which even though “clean” can be full of staph bacteria), put a loose bandage over it, give it an hour to scab over, and then apply a brand-new bandage with the ointment. Squeeze the ointment onto the sterile bandage, not onto the germ-laden wound, so as to avoid contaminating the tube’s opening with bacteria.
When my kids were of an age to ingest possibly poisonous substances for which the remedy was “Induce vomiting”, I kept a vial of syrup of ipecac in there, too. However, I see that this is now no longer the Done Thing, per the American Academy of Pediatrics. We progress, I guess.