No matter how difficult the job is, it is important to do the job safely. While some basics can be taught, every individual job requires judgement on the part of each individual involved.
Safety Glasses & Goggles
Protecting your eyes is vital in all jobs. You never know when something is going to fall off a vehicle, or fly off toward your face. Some repairs may require additional eye protection, such as a full-face shield. For proper protection, make sure you get ANSI Z87.1 Safety Glasses/Goggles, which have been tested to help ensure personal eye and face protection devices provide the necessary protection from impact, non-ionizing radiation, and liquid splash exposures. ANSI is an acronym for the American National Standards Institute, a nonprofit testing standards organization.
Gloves protect your hands to varying degrees, depending on the type of glove. Gloves are available in medium-term use (the life span varies based on how they are used and maintained) and one-time use, and which kind is used will be based on both the job at hand and the user's preference. Nitrile disposable work gloves provide good protection against chemicals, while leather work gloves provide decent short-term heat protection if they are made from thick material.
Air & Breathing Masks
Masks protect your lungs from airborne dust and debris, and some advanced ones can protect you from some chemicals as well.
Filter NIOSH Ratings
Keep hair, especially long hair, retained by a hair net or hat so that it can't be caught by rotating parts, dipped into chemicals, or touch hot components. Even beyond safety, keeping hair safe will be more convenient for you while you are working on a vehicle.
Working on a vehicle by yourself is sometimes necessary, but it is much safer to work on one with at least one other person. This person can assist or get help if you end up unable to do so yourself.
Don't wear necklaces, rings, watches, or other jewelry while working on a vehicle. These items can get caught on parts, including moving parts, and prevent escape from a dangerous situation, or create a dangerous situation where there otherwise wouldn't be one. Metal jewelry can also conduct heat, and cause burns before you can get them off of your skin.
Being Safe With a Vehicle
Keeping the vehicle safe while working on it is another vital step to getting a repair done safely.
Properly keeping a vehicle in the air will prevent it from falling on you. Jack stands should be rated to hold more than the weight that will be resting on it, and be placed on a strong, solid part of the vehicle.
Don't open a hot cooling system! The liquid under pressure can be beyond atmospheric boiling point well after the engine has been shut off, and will turn to steam as it exists the system. The steam will burn you quickly. You can't put a pressurized coolant cap back on while the pressure is escaping.
Be careful around hot oil; it will stick to skin and cause burns. Used motor oil is also known to cause cancer in lab tests.
The various fluids found in vehicles should be treated as chemicals that are dangerous to human health, and assumed to be hot.
Keep your hands away from spinning/rotating parts.
Safety relies on you knowing what is unsafe and avoiding situations where unsafe conditions happen. Working on vehicles will always have some danger involved, and it can never be completely eliminated. If a situation seems unsafe and you don't have the proper gear or knowlege to make it a safe situation, do not attempt it anyway, leave it to a professional. Repair costs are always less expensive than medical costs, and especially your life.
"Don't just think safety first, always think safety!" - Steve