If the piece is heavily tarnished or the above doesn't work, you can place the piece in vinegar and let it soak for 30-60 minutes and then scrub it with an old toothbrush. Once clean, rinse it with water. This cleaning method is more aggressive and may leave the piece more "raw" looking - it won't have as warm of color until the patina rebuilds.
Oils from your skin and the oxygen in the air are what accelerate tarnishing. If you want to slow the tarnishing process, wipe the piece with a soft cotton rag after wearing and store in a plastic bag with an anti-tarnishing strip.