DIY Silver Polish
Before polishing, try washing tarnished silver with warm water and gentle dishwashing liquid and buffing dry with a soft cloth; for light tarnish, this might be sufficient.
A frequently recommended, nontoxic trick is to fill an aluminum pan (or one lined with aluminum foil) with hot water, add salt and baking soda, and stir to dissolve. When you add the silver pieces, a chemical reaction occurs, removing tarnish. It's important to note that with this technique, the good tarnish (a desirable patina and the dark crevices in a pattern) may be removed as well, and pitting may result.
Experts recommend using a good-quality commercial polish, and there are some less-toxic ones available.