Gems lost their sparkle?

Never fear, chances are they simply need cleaning.  The trick is to access the underside of the stone and remove any buildup from it.  Diamonds can be safely cleaned in ultrasonic or steam cleaners, but many colored stones can be damaged in these automated cleaners.  For that reason I recommend hand cleaning of colored gems.  You can keep what you need near the sink and clean them whenever you feel like.

You'll want:

Mesh kitchen strainer: Put this below the faucet in the sink prior to washing your jewelry.  Soapy hands are slippery and jewelry is small; you want dropped pieces falling into the strainer and not down the drain.  This simple step will save sanity, hair, and money ;)  Plastic is better than metal, to avoid scratching your jewelry.

Mild, scum-free soap:  I use handmade soap purchased from artists on etsy.com but any nice natural soap should do.

Soft paintbrush with bristles capable of reaching into the setting and under the stone: Some people use a toothbrush, but they can dull highly polished finishes.  Go with a nice paintbrush from the craft or art supply store.

Warm water: Wash with warm water, soap, fingers, and your paintbrush, paying special attention to those hard to reach places under the stone.  Let dry naturally or use a hair dryer set on low to dry the piece without water spots.  If you use a hair dryer, be careful to avoid overheating the stone or blowing your jewelry across the counter to places unknown.

Optional, for tarnished sterling silver:

Haggerty Silversmith's Wash: Safe for just about all stones except for fine pearls, and anything with nooks and crannies you won't be able to wash the paste out of.  Mild and effective.  Use by rubbing the piece with paste using your fingers until the paste turns black with removed tarnish.  Rinse and repeat.  Use your paintbrush to get under gems, and make sure you rinse off all residue when finished.

Sterling silver turned yellow or gray?

It's an unfortunate fact of life: silver tarnishes. Tarnish first appears as a yellowish tint, then shifts to gray and black. Some people like the "antiqued" look of tarnished silver, and in fact some jewelry designers deliberately apply a heavy tarnish to their pieces. You may opt to allow your jewelry to tarnish if you simply don't wish to bother maintaining it, or if you like that look.

Removing tarnish from sterling silver: 

Silver polishing cloths are effective at removing tarnish, especially if used before the tarnish becomes too heavy. They are most useful on smooth silver; they lose their effectiveness on textured surfaces or small silver components. Chemical tarnish removers can be quite effective, but many need to be avoided or used with care around gemstones, pearls, and other jewelry components. I use Hagerty Silversmith's Wash on my sterling silver and silver/gemstone jewelry. It is much quickler and easier to use than a polishing cloth, and it's amazingly effective at removing tarnish quickly and easily. It also leaves Swarovski crystal with a lovely sparkle. It is relatively safe to use on jewelry containing gemstones, but repeated use can dull some stones, so when possible keep the wash off your stones, and wash it off quickly and completely.

Tarnish is caused by sulphur. Sources include sweat, wool, felt, eggs, leather, latex, and various other sources, so limiting exposure to them can help. Storing your silver in an airtight container with activated charcoal or commercial anti-tarnish strips is a good way to prevent tarnish when storing your jewelry.