Keeping coins safe and in good condition is just as important as finding and/or buying them in the first place. Most of the time, coins aren't just a straight piece of a metal. They are nearly always a combination of multiple metals (the amount of metals like gold and silver can affect the overall value of the coin) and are composed a way that might be hard to maintain. For example, pennies aren't straight pieces of copper, nor are they straight pieces of a combination. Pennies starting from 1982 are actually a zinc core with a covering of copper. Meaning, if mix an acid that only corrodes zinc, you will be left with a shell of copper. This change was made because of the varying prices of zinc and copper.
One of the most important things to remember is to not wash coins too many times. Really the only time a wash is appropriate is when you give a light wash when first examining the coin, as circulated coins have passed through many hands and come in contact with a large variety of different objects and places. Not surprisingly, many diseases can be transferred by coins if care is not taken. Washing coins excessively will slowly wear away at the outside, subsequently rubbing away at the design as well. Most of the time, rusts, tarnishes, and other changes to the metal itself will not be able to be washed off, and any attempts to do so will be detrimental to the coin itself. If anything seems to be washable, it is probably removable without the use of water.
To store your coins, you can either make your own display, which at best will allow contact with mere tape and cardboard, or you can purchase coin folders. Coin sleeves are also designed to keep coins safe and sound. These guys are made of vinyl and also make great coin displays. When handling coins, be careful, as excessive contact will damage the coin. It is suggested that you wear cotton gloves while handling coins and keep any contact to the coin limited to softer surfaces. Rougher surfaces will obviously scratch the coin (don't want this).
In general, keep your coins exactly the way that they looked when you first received them. Chances are that any sort of change will not be a good one (it doesn't seem possible that you can improve the state of a coin). Make sure any surface that comes in contact with the coin is softer. If you're not sure, you can try to scratch a coin that has little or no extra value beyond face value; the metallic composition should have no drastic deviation from the coin at stake.