Soak the penny in a mild acid such as white vinegar or lemon juice. A carbonated beverage can work also. After a few minutes, remove the penny and wipe it dry with a paper towel.
Rub the penny with ketchup, toothpaste, a paste of baking soda and water or brass cleaner. You can use an old toothbrush or your fingers for rubbing.
Add salt to white vinegar to shine pennies that have a lot of oxidation. This mixture won't make a paste, but you can use the edge of a paper towel or washcloth to rub the vinegar and salt on the penny.
Buy a commercial product to shine your pennies. These products are usually harsher than household products, and you will have to wear protective gloves and protect your work surface.
Use a pencil eraser to rub the oxidized layer from the penny.