Drill presses are easy to operate, but things can go wrong in an instant. Some safety precautions are obvious--keeping clothing and hair away from the bit is common sense. Other dangers are not so clear--gloves seem practical around hot metal and sharp splinters, but they wrap around a bit as tightly as a shirtsleeve will.
If you wear long-sleeved shirts, roll them up above the elbow. Don't wear gloves, rings or watches. Eye protection always makes sense--bits throw wood chips that can be painful if lodged in an eye. Metal shavings in an eye can be a serious medical problem. In other places the curls of sharp metal that spiral off a drill bit can be less serious, but a buttoned-down collar and a stiff shop apron neatly tied back can keep them out of shirt and pants.
When boring holes larger than 1/2 inch in diameter, use clamps or a drill press vise. Don't just hold the workpiece in place with your hand--the machine can tear it out of your grip. If the workpiece spins, the bit can break and the work goes sailing. Holding a piece by hand may seem faster, but it's much less accurate. Vibration can shift the work and cause the bit to wander.
Make sure the power switch is in reach. If things go wrong, you need access to it with either hand. Don't place the bit against the work before powering up--the bit could bite and immediately jam. Keep the drill press table clean. Debris underneath the work can make it tip under pressure, jamming the bit. Make sure all guards are in place. Closed motor covers keep your hands out of the drive belts.
Use the lowest speed that does efficient work. If a bit is sharp, it will cut best at a lower speed where the machine has more torque. Only light pressure is required. If a bit stops cutting, that's a sign you need to sharpen the bit. Don't bore all the way through the work on one pass. Drill a short section and back out to clear the shavings. Slow down as you approach the bottom of a through hole--the last turn is the one most likely to break the bit. A backup board makes the exit hole cleaner.
In either wood or metal, lubricating the drill bit will prevent problems. Lubrication keeps the cutting edge sharp, reduces binding and clears shavings more efficiently. Use a cutting oil in working with metal--an oil designed for the purpose won't boil off so quickly. If you see the oil smoking, back off and let the bit cool. In wood, oil leaves stains that are almost impossible to remove. Ordinary bar soap works as well and if not overused leaves very little residue.