My BIL is a vet. His rule for cat bites is the anytime the skin is broken you need to seethe doctor for antibiotics. The infection can be BAD and spread very rapidly. You can lose a lot of tissue by waiting.
This. You need to go to your doctor NOW. Any time a cat bites you, you should expect to get a horrible, nasty infection that could land you in the hospital for a few days on IV drugs. Start the antibiotics when they have a chance to work.
For anyone's future reference and for information to file away:
Cat scratches should just be cleaned out, given antibiotic cream and watched. They will usually get better on their own.
Cat bites that draw blood are an entirely different kettle of fish. The protocol I think sounds best is to wash the bite out with SOAP and water for at least 5 minutes, then douse the wound in hydrogen peroxide until it stops fizzing, rub in antibiotic cream and cover with a loose bandage. Once you've done all this, go to the doctor ASAP. If necessary go to an urgent care. Right then. DO NOT WAIT. Tell the doctor what happened; they will give you antibiotics. Take them all.
If you don't go to the doctor, expect to get a huge horrible infection and have to go to urgent care or to the ER and possible spend a day or two on an IV in the hospital.
Seriously. A cat bite is worse than a dog bite or a person bite. It doesn't matter if the cat is indoors or out, yours or someone else's. They almost always get infected. This is why you don't want to declaw your cat. If your cat gets ticked off or panics and scratches you, no big deal. If your cat gets ticked off or panics and bites you hard enough to draw blood, very big deal.