To make a dull surface shine again, start by soaking the piece in dishwashing liquid to remove dirt and grime, even overnight if it is really dirty. For an extra strong solution, add a splash of cloudy ammonia. Do not use hand soaps, bleaches or detergents, as these may actually tarnish or blacken your silver jewellery.
After cleaning, make sure you dry off the pieces properly so that droplets do not leave water marks and cause tarnishing again. Then use a special jewellery polishing cloth and rub it over the metal surface. You should immediately see the difference. Polishing cloths are impregnated with a polishing compound that will shine up the metal and leave a light greasy residue on the metal. Leave this coating on the metal as it will help delay tarnishing.
Jewellery cleaning dips can also be used to remove grime and tarnish effectively, but be sure to read the instructions on the product you use carefully. Silver jewellery with an antiqued or oxidized finish should never be dipped in jewellery cleaning products. Read here to learn how to care for pieces that has been oxidized.