Of course you can use a polishing cloth - although a bit slow with such bad tarnish. You can also use silver cream polish. However, if you don't have that handy and you have some silver plated items either sat in storage for ages or simply rediscovered at the back of a box somewhere, then toothpaste will do the job. Toothpaste is a mild abrasive, so will quickly and easily remove the worst of the tarnish with a bit of scrubbing with an old toothbrush.
So, to demonstrate:
1. Here is my as-part-of-this-project neglected salt and pepper shakers.