Pieces which are gold plated all over (gold vermeil) do not need cleaning as gold does not tarnish. Silver or part-plated pieces are best cleaned by methods which do not involve rubbing, which causes the jewellery to lose its semi-matte finish and gradually makes it become shiny. My favourite involves using hot water and bicarbonate of soda with aluminium foil in a glass or ceramic bowl. You can clean more than one piece at a time, but as this is an electrochemical process each piece needs to be in contact with the foil. It doesn't matter which way up the foil is. The hotter the water, the faster the reaction goes. Enough bicarb should be used to cover each piece. To make this method even more eco-friendly, you can re-use old foil (but it needs to be clean). (In fact, you can use the same piece of foil repeatedly.) This won’t damage the gold plating on part-plated pieces, but hot water should not be used on pieces which incorporate stones; some stones can be damaged or even destroyed by thermal shock. There are many versions of this technique available on the internet, and also many sites which have information about stones and their vulnerability/resistance to thermal shock.
The first two images show a pendant before and after cleaning by this method.
The next series of images shows a more heavily tarnished brooch. The images are taken at 20 minute intervals. As you can see, the brooch never becomes as clean and bright as the example above.
This technique works best if the piece is not too heavily tarnished.
Storing your jewellery properly can help retard or prevent tarnishing. Direct sunlight, humidity, chlorine and hairspray all accelerate tarnishing. Storing the jewellery in a box in a dry room (not the bathroom, for example) will help keep it looking bright and fresh. The jewellery should not be worn to a swimming pool, and hairspray should be applied before putting jewellery on. Salt (including from perspiration) can also react with silver, so it should not be worn for swimming in the sea.
If you don’t mind losing the surface “bloom”, you can use a commercially available cleaner such as Goddard’s foaming paste, or even toothpaste, with a soft toothbrush.
Much of this information is taken from Masamitsu Inaba's article Tarnishing of Silver: A Short Review in the Victoria and Albert Museum's Conservation Journal, January 1996 Issue 18. Many thanks.