While the basic equipment setup is that a cord is bound to your ankles and tied around your legs, wearing a full body covering safety harness can provide that extra bit of reassurance.
Know your state's laws pertaining to bungee jumping. For example, many states require a bungee jumper to jump from some kind of platform or structure that doesn't move. However, other states may allow you to jump from something moving. Research the specific laws in your state before selecting a jumping facility.
All equipment should be checked while putting it on, while waiting to jump and again on the jumping platform before the jump takes place.
The length and elasticity of the cord is important. Make sure your height and weight have been measured and that the cord has been adjusted accordingly.
Watch a specialized training video at the bungee jumping facility before each jump. If the site managers do not offer to show you the video, request to see it.