Keep the corners on your ax or hatchet handles trimmed to avoid the handle splitting. Also, a broken handle can be burned out of an ax or hatchet head as illustrated. Burning out the wood makes placing a new handle much easier in the ax head. Splintered handles are dangerous to all users and hand injuries are common.
The illustration above shows how to make a replacement handle for your ax or hatchet. Be sure to use the wedge to hold the handle in the head securely. Without the wedge, the handle will eventually start to wobble and may fly off. Your new handle may not be perfectly formed, which is difficult to do in the field, but it'll work well enough until you can replace it with a commercial one.
Remove all burrs and nicks in your blades by first using a file and then a wet stone. A sharp blade is safer than a dull one. A dull blade will often bounce when striking wood, where a sharp blade with bite into the wood. Keep your blade free of all burrs and nicks so it requires less energy to use and is safer.
A blade should have the shape of the last blade above (C), A is too pointed and B is too blunt. A properly formed blade will make cutting easier. A, by being too pointed will not cut well and you'll find it frequently sticks in the wood to the point it's difficult to remove. B, will usually bounce a lot as you strike wood and fail to bite. C, will do the job and it's what you want for a blade taper, because it bites, helps split the wood, and is the best overall shape for your use.