The leading cause of injures and fire from gas grills is leaking fuel lines. Improperly connected hoses, cracked or broken hoses, misaligned venturi tubes can release unlit propane that can quickly build up and cause an explosion. Modern gas grills are vented to prevent gases from building up inside cabinets so a slow leak doesn't pose much of a danger, but turning off the gas at the source (in this case the propane tank) is always the safest strategy.
When it comes to out-of-control gas grill fires, identify the source of the fire. If the fire is in the grill itself then carefully turn of the control knobs and let the fire die down. If the fire is under the grill and you can get to the fuel tank, turn off the tank. This should kill the fire almost immediately. If it does not, or if you cannot get to the tank valve, get away from the grill and call the fire department.
Charcoal presents its own risk due to carbon monoxide poisoning. Burning charcoal produces a lot of this gas. There were over 20 deaths in the United States last year alone from carbon monoxide poisoning associated with charcoal grilling.
The leading cause of injury related to the use of lighter fluids, is attempting to relight charcoal. Pouring lighter fluid onto hot coals causes the fluid to quickly vaporize. These vapors become extremely flammable. Without a strong wind the explosive vapors will not dissipate and will wait around for you to light the match.