Goddards Silver Pad (formerly known as Silver Foam) is like a pink soap that you rub on with a wet sponge (supplied). It contains a chemical that breaks down the tarnish. The tarnish comes away in the lather, you rinse it off, then polish with an old vest a soft cotton cloth. You can then wash in warm water and WUL until they get tarnished again. John Lewis have an own-brand which appears to be the same.
With silver plate in particular, you must not use anything abrasive such as toothpaste, as you will rub away the thin layer of silver.
When you are polishing your solid silver, hold it with your thumb on the hallmark so it doesn't get worn.