Silver tarnishes quite quickly when it comes into contact with oxygen or sulfur. Happily, the best measure to prevent this is wearing your silver jewelry often. The oils in your skin help prevent tarnish. (This varies depending on your own body chemistry.)
When storing your silver jewelry, use an airtight plastic bag (like a Ziploc) and if you can, put in a piece of chalk or a silica gel (the little packets in new shoes) - these will help prevent tarnish while you are not wearing your silver.
If your silver is tarnished the best way to clean it is with a jewelry polishing cloth. (Here is an example, but they are widely available!)
There are also many viable home remedies for cleaning silver online. Just remember that silver is very soft so use something like a very soft baby toothbrush to avoid scratching.