LPT: If you have a stainless steel mug (Contigo, Yeti etc) use Denture Cleaning Tablets (Polident) to remove coffee/tea stains.

Fill your mug with hot water and pop a tablet in and let it soak overnight. By morning you will be able to gently wipe away all the brown buildup.

Edit: This is for when regular soap does not work. Do not try scrubbing stainless steel or use abraisives. If the surface becomes scratched you will end up with even more gunk/stains.

Edit 2: This is for when vinegar, dishwasher soap, baking soda and other methods aren't working. This is a method that involves very little effort.

Polident tablets are relatively cheap. It's less than $10 for 100 tabs (in Canada). You wouldn't use them daily, only when the stains get beyond conventional cleaning methods can handle.

Also - Bleach is bad for stainless steel.