What is the best way to clean gold jewellery of tarnish?

Gold does not tarnish period. You probably have an oily film on your jewelry that has dried and discolored. The household detergent should work fine. Also try soaking everything in alcohol to remove the oil from your body. After all of this to shine everything up, use toothpaste on an old toothbrush.. Scrub and rinse to get all the little crevasses.