Hydrogen peroxide is cheap, bio-compatible, and kills nearly any anerobic bacteria it touches. Since I am often found in a woods, heat with wood, have animals and grow roses, it sees frequent use in my corner.
Topical wounds, bites, mystery red patches, etc. benefit from a moistened pad that stays on a few minutes. Puncture wounds are harder, but fixable: Soaking for 5+ minutes until you see bubbles does well. If a fingertip, just shove it in the bottle and upend the works. After 30 minutes immersion it will get to fingertip nerves and cause uniquely durable pain. If unable to find real medical assistance, I would consider injecting it into a deep wound, but the cleansing might come at the price of considerable cell damage.
For thorns, splinters, etc, a razor blade, needle or xacto type knife is helpful to open a pathway, then peroxide to taste. Quick and thorough initial treatment is better than remedial care, but you do what you can when opportunity permits.
Peroxide is really reliable and kills from inside out. If redness remains after 12H, open the wound a little and repeat. Doesn't take much to treat normal small wounds. I carry some in my travel kit in a hotel shampoo-size bottle. Wears out for travel after a month if not very tightly sealed.
If no peroxide handy, then strong salt solution in water or water alcohol. Hurts.. but it works. Sucks the juices right out of the 'teria.