If it's gold and has gemstones (diamonds, sapphires, etc) you can buy a small jar of jewelry cleaner and put the jewelry in it for a few minutes then brush gently with the small brush provided. I've even put it in a small glass of ammonia and water. Do not use this for pearls or opals. Silver jewelry can be cleaned with a sliver-cleaning cloth.
Gold? Silver? Faux? Gemstones? Pearls? Opals?
Here are a few methods depending what you have.
For gold, silver, and gemstones I use white toothpaste and a soft toothbrush. Rinse gently and dry with a microfiber cloth. If it is costume jewelry I try to avoid water if there is any paste to hold in the stones.
Lots of tips:
I sell a lot of jewelry on Ebay. I use white toothpaste (not gel) and a damp toothbrush. Also works on copper, brass, gold, silver, porcelain, ceramics, sneakers, plastics, etc. For my sales I needed one product for cleaning everything.
I have an ultrasonic jewelry cleaner it was very inexpensive
here is how to clean jewelry to make it shine like brand new! Soak your pure silver bracelets, rings, and other jewelry in a mixture of 1/2 cup white vinegar and 2 tablespoons baking soda for two to three hours. Rinse them under cold water and dry thoroughly with a soft cloth.
A friend of mine who sells jewelry swears by using a soft toothbrush dipped in white vinegar and buffed with a soft t-shirt.