The most obvious method, soap and water, is not what I choose. Soap can leave a residue in the pot that will affect the flavor of your tea. Fortunately, an easy alternative is probably already in your kitchen cupboard: baking soda.
Rinse the teapot, then shake some baking soda into it (a tablespoon or so), making sure to get some on the rim. With a damp cloth or sponge, gently rub the soda on stained areas, including the underside of the spout and inside the pot. This should lift all but the most stubborn stains, and not harm any gold trim.
This works for china, too – and remember to do the teapot’s lid as well!
I have used baking soda for years and it works just great! Though I usually rub it around with my fingers — easier to get into the crevices and such. You can also make a paste of baking soda and water and rub it in rather than sprinkling.