I just filled the mug, like I do with mugs and cups now with diluted bleach and water for an hour or so, which makes tea and coffee stains disappear. You can also use a wire scourer and washing up liquid if you don't like using bleach.
My wife won;t have bleach anywhere near things used to contain food
and she reckons that using anything scouring just cleans it but adds fine lines that make it stain faster afterwards
She is certainly right about the second one - I have only used a cloth on my current mugs and they are lasting much longer
The first one I am not so sure about - I reckon and damn good rinse afterwards is fine
(I have a Chemistry degree - I am qualified to test for any residue but apparently what her Mum said ranks higher than a degree from a good UK University!!!)
BTW - I drink black coffee (quite strong) and tea that I like to brew for a while - so I am quite capable of staining a mug!