Dropping the chain normally does not happen when the cables are properly adjusted. When you first get a new bike, the cables will stretch, so you should take the bike into the LBS to have it adjusted. Periodic adjustments are always needed the more wear you put on the drivetrain.

There is a fitting you can buy which fits on the botttom of your seat tube and it will prevent the chain from dropping off the inner chainring. It is called a "Dog Fang Chain Guide" and it is made my Deda; cost, $14.99. Most LBS's should have it or they can order it for you. Although some people think it is "Fred", including my LBS, I have used it on 2 of my bikes. It is a small hard rubber ring with a hook (dog fang), which fits close to the innner chainring. If the chain should drop, you just keep pedalling and the chain goes right back on. It is so small and inconspicuous, you don't even know it's there.