Before chopping any wood, take a gentle practice swing to check that the axe will not catch on anything (such as an overhead branch).
Consider what will happen if you miss whatever you are chopping at - will you accidentally hit a finger? A foot? A bystander? Leave plenty of margin for error.
Make sure the axe head is firmly attached to the axe handle. If it is loose, you may tighten it by wetting the handle, by driving a hardwood wedge into the handle through the eye of the blade, or by rapping the axe handle vertically on a hard surface.
Keep bystanders away by one arms length plus two axe-lengths.
When handing someone an axe, present the handle to them rather than the blade.
Walk with the blade facing away from you.
Sheathe the axe when it is not in use.
Always maintain firm footing when using or carrying an axe.
Stop when you are tired and rest. Tired people are more prone to accidents and mistakes.