When you hold or handle your coins, the number one rule to remember is to never touch the faces of the coin. Always handle coins by the edges. If the coins are in Mint Set packaging or in plastic capsules or holders, you should never remove them unless absolutely necessary, especially if the coins are in grading service holders or capsules from the U.S. Mint.
Sometimes, though, you may want to remove a coin from its capsule to examine it more closely or photograph it. Here are some tips and things to keep in mind so that when you handle your coins, you do minimal damage to them.
How to Hold Loose Coins
If the coin is loose (not in a capsule or plastic holder) your main concern is preventing damage to the coin. Follow these steps:
Establish a safe working surface. Lay a soft towel or other fairly thick but soft cloth on the table, so if you drop the coin, you don't end up with a ding to edge, or worse, a rolling coin that winds up on the (dirty) floor. A soft cloth will cushion the fall and prevent rolling, so always work over a cloth or specially-made felt-lined coin tray. Be careful about using a towel that smells heavy with fabric softener, though, since this chemical could hurt your coins.
Make sure your hands are clean. Although some people advocate wearing cotton gloves when you handle your coins, I don't recommend them (and you won't see most professional coin dealers using them, either!) Wearing gloves makes you lose tactile sensitivity in your fingers and you become more clumsy, thereby risking a dropped coin. If your hands are clean, you don't need to worry about damaging the coin from oil or dirt anyway so gloves are an unnecessary complication.
The primary way that coins are damaged through handling is when someone touches the surface of the coin and gets skin oils and/or fingerprints on it. Although in the past, experts recommended that you wash your hands before handling your coins, I think even this step is unnecessary today, since the primary offender is skin oils anyway. Just use one of those hand sanitizers before handling your coins, paying particular attention to your fingertips. Hand sanitizers are alcohol-based, which very effectively strips the oil out of your skin while cleaning your hands at the same time. Be sure to use unscented sanitizer, since the scented kinds might leave behind other trace perfume chemicals that could get on the coins.
Handle the coin by the edges only. There is never any good reason to touch the surface of a coin with your fingertips. Even when your fingers are clean, any little bit of dirt, oil, or grit can damage the surface. Pick the coin up with your primary hand (most people are right-handed), using your thumb and forefinger to hold the coin securely. Always keep the coin above your soft cloth or tray, leaning in towards the coin rather than drawing the coin towards your face. Be sure to keep the coin away from your breath-stream, too. (See the next tip for details.)
If you are looking at a loose coin at a coin show or at a dealer's shop, be aware of any felt-lined coin tray or other soft surface the dealer may provide, and be sure to hold the coin over it when you pick it up. Most dealers put their coins in some kind of holder, at least a cardboard 2x2, but from time-to-time you may have to hold a loose coin without any drop protection. In this case, use your other hand as a "catch-guard," holding your cupped left hand slightly below the coin so that if you do drop it, your other hand will catch it.
Never under any circumstances touch the surface of someone else's coin, even if the coin is a crappy, well-circulated, low-cost specimen in someone's junk box. Having respect for other people's coins is the number one point of etiquette to observe at coin shows, coin clubs, and in dealer shops.
Do not spit on your coins. As obvious as this seems to be, I see people spitting on coins all the time at coin shows. When you talk, little tiny flecks of saliva spew forth from your mouth, some of which can land on unprotected coins. This is often the cause of spotting that you see on some copper, silver, and Proof coins. When you hold a loose coin, keep it well away from your breath-stream. Although breathing on it probably won't damage it (it might depend on what you've been drinking), oftentimes people are chatting while looking at coins, so make it a habit to hold the coin out of your flow of breath (and spittle.)
Handling Proof Coins
Unlike regular business strike coins, especially ones that have already been circulated, Proof coins present a special challenge for who people decide to handle them outside their capsules or protective packaging. Because most modern Proof coins have pristine, mirror-like surfaces, any tiny little bitty imperfection is a disaster for these coins. Special care must be taken if you feel you have to handle them.
Establish a sterile workspace I don't mean "sterile" in the medical sense, but you do need a very clean, lint-free, dust-free space. Make sure your hands and fingers are very clean, and have everything ready in advance for whatever you plan to do. If you're photographing the coin, have the camera set up and ready. If you're examining the coin, have your microscope or loupe or magnifier close at hand with the lighting already established. If you're planning to move the coin from Mint packaging to a capsule, have the capsule open and ready. You want to minimize the time that the Proof coin is exposed to the open environment.
Hold your breath. Okay, literally holding your breath is probably a bad idea, but don't breath on the coin, and especially do not talk or chew gum at all while the Proof coin is exposed to the air. Make the exposure as brief and safe as possible. Proof coins have very fragile surfaces, and you will lose several grade points if the surface gets damaged by something that eats into or scratches it.
Coin Handling Summary
Handling coins doesn't need to be a frightening experience. Just use common sense, clean hands, and a soft working surface and you'll do just fine. If you have any questions, come ask them in the Coin Collecting Forum.