This is a photo of my new favourite cup (which I bought in Disneyland). It has been in use for a few months and has acquired a stain from the tea I drink.
So, how do you remove that yucky stain? Easily. Use baking soda. Dampen a cloth and dip it into a small amount of baking soda. Rub all around the cup. Rinse and admire your good as new cup.