Always keep your jewelry in a cool, dry area. If you can, keep it in a sealed plastic bag and refrain from touching it with makeup, oils or perfume on your hands. All of these will lead to quicker tarnishing.
When you first buy your pieces, a quick coat of clear nail polish will help create a seal. If you choose not to paint your pieces-wipe down with a paper towel each time you wear them. This removes any dirt, oil and residue that builds after each wear. Do not use any kind of cleaner as these are intended for actual gold or silver; doing so will discolor your jewelry.
If you want to get your ornate gem pieces clean, use a soft toothbrush and warm soapy water. Gently brush and dry off with a paper cloth. Leaving any trace of water may cause rust.