There are a number of simple steps you can take to get the best use from your chainsaw.
1) Wear proper safety apparel. Think safety first. I recommend wearing eye protection, hearing protection and leg protection. Chaps or fallers pants are not always necessary but for a small investment they might save you from a serious injury. Next make sure your saws safety features are working, most importantly your saws chain break mechanism. It is vital that you work with a spotter or a partner when cutting wood, especially when doing pruning or when the work involves climbing a tree or a ladder. If a particular job is making you nervous or uncomfortable it is probably best to call in a professional. Even a moderate sized branch can easily injure or even kill you.
2) Make sure that you mix your fuel properly. Most saws manufactured in the last 10 years will run properly on a mix of 50:1 or in some cases more like 40:1 although that is typically on low end chain saws like Ryobi, Homelite, MTD, Yardworks etc. If you own one of these, I would recommend you find a way to replace it sooner than later with something decent from a manufacturer like Stihl or other high quality brands.
3) Use chain oil. Although it seems obvious, I have had cases where a customer is using 10W30 or similar for bar & chain oil or even used motor oil which, by the way will clog up the oil lines and damage your oil pump.
4) Use clean, fresh fuel and use a fuel stabilizer. If you let your fuel sit around or in your saws gas tank for more than a couple of months, the fuel wil break down and cost you plenty in repairs from varnish buildup that will occur in the carburetor and fuel tank. Using a good fuel stabilizer is more or less a must these days. Make sure you replace your spark plug at least once every year or two. Check your air filter regularly and clean or replace as needed. If your air filter is clogged with sawdust or dirt or water the saw will not run properly.
5) The most significant factor in the performance of your chain saw is the chain. Make sure it is always kept razor sharp and is filed properly. If you have a dull chain, which may be indicated by sawdust instead of chips dropping on the ground while cutting or the bar getting hot or when you are having to force the saw into the wood to make it cut, replace the chain or sharpen it. I always recommend having at least one or more spare chains on the job site so you can quickly get back to work when your first chain gets dull. A chain will easily lose an edge when cutting near or on the ground or cutting dirty wood or wood that may have dirt or nails in it. A dull chain will cause your saw to work much harder than it was designed for, it will cause you to work harder than necessary which often leads to injury, and it may make you frustrated by not performing as you expected.