Serious practitioners of numismatics (the study and collection of coins, paper money and medals) will tell you that you shouldn't clean your coin collection because collectors prefer that "natural look." But, hey, they're your coins and if you want them shiny, here's how to get them that way with minimal damage.
Handle coins by the edges to avoid putting fingerprints on the flat surfaces.
Avoid silver and other metal polishes - even jewelry polishes. They are much too harsh and will leave tiny scratches on the coin's surfaces.
Soak the coins in olive oil or soapy water for a few days and then rinse them well with clean water.
Try non-abrasive cleaners such as denatured or isopropyl alcohol, acetone, or an ultrasonic jewelry cleaner (with water) if the soaking doesn't do the trick.
Blow dry or pat cleaned coins with a soft cloth, not tissues or paper towels.
Never rub the coin - unless it's your good luck charm.
Be extremely careful if your coins are old. If you mess up, you could seriously decrease the coins' value. Consider taking them to a professional. Tarnished coins are better left as they are. Collectors call it "toning" and it's a desirable look.