Silver is oxidized in air to silver sulfide easily. To avoid it, you can store your jewelry in an airtight place. As to how to get ride of the tarnish, you can try place the silver chain in a disposable aluminum pan and pour boiling water in it. To which Baking soda (0.25 cup per liter of water) is then added.  You should see the tarnish start to disappear quickly (I hope...) 

BTW, I never try it myself. But it sounds reasonable to do from a chemist's point of view.