Your two main priorities when storing your gold and silver are to avoid tarnishing/dulling and avoid scratching. To avoid tarnishing you will need to reduce the amount of moisture in the air. Store each piece of jewelry in a separate cotton or jewelry bag. The bag needs to be able to breathe, if the bag is air tight it may sweat which will cause tarnish.

Good quality Jewelry bags are usually treated with chemicals that reduce tarnish. A good quality jewelry bag will save you a lot of trouble in the long run. The pieces should be stored individually, if stored together the pieces will scratch each other. If available place a moisture absorbing product inside your jewelry box. White chalk, charcoal or silica gel will absorb the moisture which causes tarnish. 

At last, I would like to give you a rough guideline to prolong the life of your Jewelry

Apply makeup and perfume before putting on your jewelry. The chemicals will damage sensitive gemstones (like pearls) and tarnish metals like silver.

Remove jewelry when cleaning. Ammonia, detergents and other household chemicals will damage jewelry. 

Do not shower with your jewelry on for the same reasons as the above point. In addition to damage from chemicals any jewelry which falls off (soap will loosen rings) you will lose your jewelry to the plug hole.

Do not swim at the beach or swimming pool with jewelry on. Chlorine and salt will damage your gemstones and metals.
Gold and silver are soft metals and will bend easily. Avoid doing anything strenuous with gold and silver jewelry on.