People have been accidentally cutting themselves for as long as there has been something out there to cut ourselves with. Take a look at the back of your hands and chances are there's a scar there from a cut you can't even remember how you got. Not clumsy or even accident-prone, adults probably use more bandaids than they do sharp objects.
Infection, by definition, is an invasion by a disease-causing organism into a host's (your's) bodily tissue. This becomes all the more substantial when there is a tear or break in the skin, which would otherwise do a near-flawless job at keeping malicious organisms out. Cleaning and caring for a laceration should never be overlooked as the result could be an infection.
Here's a simple, straight forward guide on how to address and disinfect a minor wound. (given that you have time to go through all 4 steps, and are not bleeding profusely)
If you notice that your cut is severe in nature, please contact your Doctor immediately.
Step 1: Stop the Bleeding
Cover the wound with a clean cloth/bandage to stop or limit blood loss. If possible, elevate the wound above heart level, so less blood flows to the injury.
If the wound does not seem to stop bleeding, call your doctor. The cut may be deeper than you think, and stitches may be required.
Step 2: Gentle Wash
Using lukewarm and clean water, rinse the wound. Soap may irritate the cut, so try to stick with just water. Washing the injury greatly reduces the risk of infection but also helps in removing any dirt or debris that may have been lodged.
If you notice lingering pieces of debris after washing, call your Doctor.
Step 3: Apply an Antibiotic
After washing and drying, apply a thin layer of antibiotic ointment cream such as Neosporin or Polysporin to the wounded area. These creams help fight infection and also create a physical barrier to keep germs out.
If you fear you may be allergic to antibiotic creams, simple vaseline serves as a great replacement. Although it is not in it's nature an antibiotic, vaseline will provide a moist environment ideal for healing.
Step 4: Dressing
Place a bandaid (or any type of variant, such as gauze and medical tape or tieable clean cloth) over the entire lacerated area. The dressing serves to create a physical barrier keeping germs out but also keeps the wound moist, which greatly helps in the healing process.
Redress wound as needed or when it becomes wet or dirty.
Remember: a clean wound is a happy wound, one that will greatly limit infection and scarring.