The most important part of cycling safety is wearing a correct fitting helmet. A helmet will not protect you entirely though and riding safely is the best form of protection.
Gloves are an important part of your cycle safety gear as instinctively when falling off your hands go out to break your fall. Other protective equipment is available ranging from knee and elbow pads to full body armor.

Road safety tips.

Make sure you are seen. Wear bright clothes, ride with lights and ensure that cars can see you at all times.

Ride well away from curbs and parked cars. An opened door in your path is a very real danger when riding in an urban environment. 

Don’t swerve in and out of parked cars and obstacles keep a predictable steady course.

Make sure other road users are aware of any maneuvers you plan to take, look and signal before you do anything. If in doubt stop safely and perform your maneuver on foot.

Off road safety

Never ride beyond your abilities, if you are unsure of a trail section complete it on foot.

Know the trail. If you can, walk a trail first, taking in all obstacles and potential dangers.

Keep your speed in check. Always make sure you are able to slow and stop quickly if the need arises.

Tackle blind corners with extreme caution, you never know what is waiting around them.

Make sure your equipment is up to the job, slick, thin tires will not be suited to rough and wet off road conditions.