How do I take care of a tooth infection without going to the dentist?

You really can't. If your tooth is infected i would see a dentist right away. Antibiotics may make it feel better temporarily, but it will come back, and having an infection anywhere, especially in your head.
Get a string. Just kidding. You do not want to treat this yourself or pull the tooth out. Go see a dentist.
Too little too late. You likely need antibiotics if you have a tooth infection. You need to see a dentist. Good dental hygiene can help keep this from happening in the future. If you wait, this could become a much larger issue requiring more extensive dental work which could end up being more painful and more costly financially so go get it taken care of now.