What is the best way to clean stainless steel kitchen appliances?

Use a nylon scrubbing pad soaked with dish-washing liquid and scrub the stained area in a circular motion until the blue or brown stain has disappeared. repeat several times if necessary to remove heavily discoloured or stubborn stains.

Once the stain is removed - rinse the item thoroughly with clean water before drying. If this approach doesn't work - then you can always use any number of commercial products designed specifically for stainless steel discolouration removal.