Salt. Seriously run hot as you can water in the teapot & then add a bunch of salt & swirl about, the salt will scour it clean. If it's copper then use a baking soda paste & scrub it. Cleans really well. For the cups use a magic eraser.
if you haven't spent a fortune on them and they aren't really "valuable" then just put them in the dishwasher. We created a tea-set for my DD's with thrift store purchases - they don't match, but they are functional and cute, and I don't care if they break them. (they haven't)