Fur future reference (since it sounds like this handled itself)…
First thing is to stop the bleeding with 15 minutes of direct pressure. I mean it! 15 minutes! Stop peeking! If you see blood on whatever you’re using to absorb and apply pressure - paper towel works well - don’t take it off, put more absorbent material on top of the bloody one. Taking the bottom one off disrupts the clotting your body’s trying to do to stop the bleeding and makes it start over.
When the bleeding has stopped, current first aid would guidelines state that wounds should be washed clean with running water. That’s it. No soap in the wound, although soap on the intact skin around the wound is okay.
If the wound can be cleaned with running water (that is, there’s no debris stuck in it) then you should put a bandage on it to keep it clean while it heals. If the wound cannot be cleaned of debris and foreign matter with running water, then you should leave it open to the air and seek medical attention.
Don’t breathe on it or blow on it or stick it in your mouth! Common reactions to small injuries, but obviously a bad idea. Mouths are full of germs!
Mayo Clinic is still recommending triple antibiotic ointment (Neosporin, et al) for home treatment of minor wounds, but that’s becoming controversial in the industry. Overuse of triple antibiotic may be contributing to the evolution of some antibiotic resistant bacteria. Research is still being done, and I don’t believe the CDC has a particular stance on it one way or the other right now.
I’m partial to a smear of honey on small wounds myself. No one with official letters is recommending it for home first aid right now, although we do have these nifty wound packing materials with “medical grade” honey available for professional use.
Wounds which don’t stop bleeding with 15 minutes of direct pressure should have medical attention. Wounds were you can see stuff inside you that you normally can’t, like fat or tendons, should be seen by a doctor. Puncture wounds (anything deeper than a pencil eraser) and gaping wounds or wounds with jagged edges should have medical attention.
Bandages should be changed once a day, unless directed otherwise by your doctor. Clean the wound when you take the old bandage off. Again, no soap in the wound, but around it is okay.
No butter. No Crisco. No alcohol. No hydrogen peroxide. No mecurochrome that’s been sitting in your mom’s medicine cabinet since 1982. No nothin’. Most uncomplicated, uninfected wounds do much better if they’re left alone, other than being kept clean, and the wound bed is kept moist while the skin around it is kept dry. For a simple kitchen cut, this most often means just a band-aid after washing.