After use give your piece of jewellery or silverware a wipe over with a soft jewellers cloth or salvet (this is what we use in the jewellery trade), these have a light polishing compound on that will remove very light scratches and take away the finger print marks from the item.
Alternatively the tarnishing is heavy and your item has yellowed or blackened you can use a silver polishing solution or cream, just make sure that you use a soft cloth to wipe it or some cotton wool as you do not want to scratch the item.