Ozidization or tarnish occurs on your silver jewelry when sulphur compounds in the atmosphere chemically react to the silver. Tarnishing can be avoided when you properly store your silver.
When you put your silver jewerly away add a silica gel pack (that often comes in shoe boxes) or caulk to absorb moisture. (Not a good idea to store your jewelry in a bathroom) You can buy bags made with silver cloth embedded with thousands of fine silver particles, which absorbs tarnish producing gases before they reach the silver. A less convenient way to store, is wrap the silver in acid free tissue paper then put in a ziplock air tight bag.
Avoid direct contact with wool, felt, plastic wrap, gasoline, motor oil, latex gloves, elastic bands, hair spray, perfume, deodorant, lotions, soaps, cleaning products, chlorine from swimming, eggs and mayonnaise.
The best way to polish your silver is with a silver polish glove or cloth. I have used a number of them and find the ones with anti tarnish elements are the best; Both Selyvt cotton cloth from the UK or Hagerty from the USA. I use Town Talk Polish anti tarnish silver foam from the UK for heavily tarnished pieces. The clothes work very well most of the time. You can purchase at a grocery or hardware store and all can be ordered online. If you use gloves while cleaning avoid ones with rubber.
I do not recommend the use of tooth paste which is sometimes given as an at home solution on advice blogs. The ingredients may damage the silver or speed up the tarnishing process. I also don't like the dipping method or harsh chemical cleaners as they can damage your jewelry (especially if they have gems or crystals) and can leave a greasy finish.
On some of the relief, textured and respousee antique and vintage silver jewelry I leave some of the oxidization (tarnish) as it really looks better to me. I don't return the finish to a high new looking polish because I like the little nicks and dings that tells a story.