Filleting a fish needs to be done carefully. Apart from a sharp knife you have sharp gill plates and sharp spines to contend with. On top of that the fish is slippery so there are many factors contributing to making the process potentially dangerous.
Never, ever cut towards yourself! It seems like this one should go without saying but people still do it.
Wear non-slip rubber boots and an apron so you can stand comfortably close to your filleting bench without worrying about getting water or guts on your clothes.
Wear a glove on your non-knife hand. This hand should do all the handling of the fish. The type of glove isn't too important but it should give you good grip while allowing you to still 'feel' the fish. If it's waterproof too that's a bonus.
Keep your knife hand and the handle of your knife clean at all times (and preferably dry). Avoid using your knife hand to handle the fish. If you do handle the fish with your knife hand, rinse and dry your hand before proceeding.
Fillet the fish on a cutting board that will help prevent the fish sliding away. Keep the board clean and wipe away blood and guts from previously cut fish before starting on a new fish.
Make the first cut down the dorsal spines from head to tail as we demonstrate in our videos. You will sometimes see people start at the tail and cut upwards to the head. Personally we don't like that method as it encourages you to cut towards your own hand and you are cutting on top of the scales which makes it easy to slip.
Remember the spines and gill plates! People are often very careful not to cut themselves with the knife but don't pay enough attention to other dangers like the sharp gill plates and the spines. In our experience cutting fish at factories we have actually seen more people injured with nasty puncture wounds from spines than cuts from a knife.
Finally, take your time and focus on doing a neat job and getting good recovery, never rush or try to go too fast.