Only cut tree limbs that you can reach from the ground. Hold the saw with both hands. Never overreach, and never cut above shoulder-level. And if you're inexperienced, cutting down trees is a job best left to the pros.
A lot of chain-saw injuries involve kickback, which is what happens when the tip of the saw contacts the wood and lurches back at you. The best way to prevent kickback is to never let the tip of the saw contact the wood or the ground. Always operate the saw to the right for better balance. Keeping the saw sharp and well oiled will also help prevent kickback. And tightening the chain is also key. The chain will loosen as you're cutting and it can come off the bar, so you need to tighten it every 10 to 15 minutes.
To stay safe, you'll want to invest in a helmet as well as in sturdy gloves and steel-tipped boots. Consumer Reports says hearing protection is also important. Sound levels for gas saws can exceed 100 decibels. Even electric saws can be loud enough to cause hearing damage. It's a good idea to look for a helmet with the hearing protection built in.