Is there anything I can add to homemade bread to preserve it?
My whole wheat bread takes 4x to go off than my white bread. Even a mixture of 50% whole wheat flour will make it last longer. But that will only help if the problem is the bread getting hard too soon. That can also be prevented by keeping it in a plastic bag. You'll get the mold before the bread goes stale.

If you're keeping the bread in a plastic bag, try paper bags.

To solve the mold problem, the traditional way is to add some acidity. For example, you can add a sourdough starter. If you don't like the taste of sourdough, a poolish starter should also help. The bacteria it grows will prevent the mold growing.

I know it's not what you're asking, but freezing will also help. When you bake more bread than you eat, freeze it wrapped in a plastic bag and thaw it overnight and you will get a good, fresh bread.