There are several ways to clean jewellery, depending on the degree of dirt.
To clean your jewellery regularly, in order to maintain it’s shine, the simplest way is to wash in in warm water and, if necessary, with some neutral soap. If you need to scrub, a soft sponge or cotton cloth are best and a cotton swab can be used to reach small areas. Some people recommend a child’s soft toothbrush but these cause small superficial scratches so they should be avoided on polished items.
To prevent oxidisation, you should keep your jewellery in a plastic bag or jewellery box with an anti-oxidisation agent – cotton paper or cloth. If the item does oxidise, it can be cleaned by specific products. At the supermarket you can find several metal cleaning products that work well in most cases. My favorite is silver foam by Town Talk. It’s a red paste that you apply with a damp sponge and it cleans silver well and makes it shine. Afterwards you should always wash it well to remove any trace of chemicals. Tap dry on a paper towel or cotton cloth before use.
I do not advise the use of silver dip or bath. It’s meant for silverware and it’s a bit harsh for jewellery, specially if it has stones.
For the occasional polish there are polishing cloths like Sunshine cloth or Town Talk’s silver or gold polishing cloth. Just rub the cloth over the metal to remove dirt and oxides and make it shine. These cloths turn dark with use but shouldn’t be washed because that removes the polishing agent they’re impregnated with and prevents them from being used again. The Duraglit cotton is another option.
If your jewellery has been treated with a patina these cleaning products should be used with care, if at all.
Some people use a solution of sodium bicarbonate (baking soda) to clean chains, since it’s so hard to get into all the links by hand. Place the chain on a tray, cover it with the sodium bicarbonate and pour hot water over it. As the solution bubbles, it cleans the silver. This should not be attempted on jewellery with stones.
A rotary or vibratory tumbler is sometimes used to clean jewellery as well but most people don’t have them at home. You place the jewellery in the tumbler along with water, soap and steel shot and let it rotate for a while. Soft stones can be severely damaged by this process and thin metal can harden and become brittle through repetition so I don’t recommend it as a constant method.
An ultrasonic machine is used by jewellers and is a very efficient cleaning method. Again, not all stones are safe, especially if they have cracks, but it’s one of my favorite methods because it’s fast. These days there are inexpensive ultrasonic machines available for home use. They don’t heat the water but you can pour previously heated water into it, with some ammonia based detergent and let it run for a couple of minutes. If the result isn’t good enough, rinse and run it again.