The enemies of Silver are Air & Moisture, so when not worn silver jewelry should be stored in a way that keeps Air & Moisture away. A Jewelry box is typically lined with a fabric that helps keep Silver nice, and a Ziplock Bag works well for traveling. If you simply keep your silver put away when not being worn there should be no need to purchase these anti-tarnish items.
If needed there are specialty items you can buy:
3M makes an anti-tarnish item that can be put into a Jewelry Box to reduce tarnish.
There are anti-tarnish fabric bags (made from a specialty fabric).
Chlorine (in pools & in cleaning products) will tend to oxidize the surface quite quickly. You should remove Silver whenever it might come into contact with chemicals (soaps, cleansers, and more). I have also discovered that some sunscreens can cause silver to oxidize, and the oily nature of some lotions & sunscreens simply gets jewelry dirty.
Silver-Only Jewelry: I found this to be the best method & it uses simple things most of us have at home.
What you need: A Glass Bowl, Aluminum Foil (must be Aluminum), Baking Soda, & a sink or dishpan to place the bowl in (for overflow).
How to clean: Line your glass bowl with Aluminum Foil, shiny side up. Place silver in the bowl making sure it touches the aluminum well (for chains you will want to wait until after you pour in the water), cover with baking soda (a few teaspoons or so). Place the bowl in the sink/dishpan, pour boiling water into the bowl - it will bubble & may overflow. You may sprinkle a little bit of salt into the bowl to speed up the process if your piece is especially dirty.
What this is doing: This is a chemical reaction that turns the Silver Oxide back into Silver.
Silver with other metals, stones, or other items:
There are a variety of polishing methods, cloths & chemicals, but they all remove silver that has oxidized rather than returning it to its original chemical form, over time some of these methods can cause pieces to become dull/scratched - so use care. Additionally, use extra care around stones as many can be scratched/dulled as well.
I like: Sunshine Cloths - fabric polishing cloths - come as a single cloth, cannot be washed but last a good long time. Connoisseur Wipes - disposable polishing cloths - come with several in a little box. Last I checked both were readily available.