Cleaning silver with store-bought silver polish is always an option. Take, for example, Hagerty's Silver Polish. It promises to remove fingerprints and protect your silver with a tarnish preventative, making your silver polishing experiences less frequent. Like most silver polishes, it is to be rubbed on with a cloth or sponge, meant to dry on the piece being polished, then rinsed off. Though probably effective, store-bought silver polish is a purchase you don't have to make. Baking soda will work just fine.
Polishing silver with baking soda can be done two ways. The first is useful for smaller pieces like silverware. The second is more effective for larger pieces like trays. If polishing smaller pieces of silver, lay a piece of tin foil (shiny side up) on the bottom of a bucket or bowl. Place the silver on the foil. Multiple pieces can be piled up on top of one another. Fill the bucket or bowl with warm water and add enough baking soda make the water fizz. Soak the pieces in the fizzing water for 30 minutes. Remove the silver from the bucket or bowl, rinse it off and dry it.
Baking soda can be used to polish larger pieces of silver, pieces too large to fit into a bucket or bowl, as well. Simply make a paste out of the baking soda and water. Then apply the paste to clean sponge or cloth and wipe the paste onto the piece of silver. Be sure to rub it into crevices. Use an old toothbrush for detailed etchings and carvings. Let the paste sit on the silver for, again, about 30 minutes. Wipe the baking soda polish off with a clean cloth or sponge soaked with water; then dry the piece off with a dry cloth. Baking soda paste that was not wiped clean will leave a white residue. If you do have residue, wet another cloth and wipe it off.