Tarn-x will cause silver to tarnish even more later. Tarn-x makes your silver smell like death and its impossible to get rid of. Tarn-x is an acid and it eats away layers of metal that's how it gets so shinny. Its horrible for coins and other silver. 
Toothpaste works good but its corrosive and can put hairline scratches on it. Electrolysis is best cause nothing but electricity touches the actual coin. However if you over cook an item it can get pitted or damaged too. You can make a lesser type of electrolysis with just foil, hot water, and baking soda. 
This works great for light tarnish and dirt. A simple really long bath in warm distilled water is one way professionals do it for heavy dirt or corrosion. That won't work on black tarnish so I'm not sure what they do for that... 
