A few observations based on personal experience and observation of others:
1. Don't ever assume that someone who has cut a lot knows what they're doing.
2. Tiredness and saws don't mix - don't cut when you're tired, don't run a tired or dull saw.
3. PPE - always wear your boots, chaps, eye, and ear protection. Wear gloves unless you're one of those people who absolutely cannot stand to wear them. If felling, seriously consider wearing a helmet; you don't need one when cutting stacked up logs in your yard, in my opinion.
4. Buy Mr. Beranek's book, Fundamentals of General Tree Work. Read it. You'll learn a lot.
4a. After reading Mr. Beranek's book, watch some of Eric's videos (member name EKKA on Arboristsite, owner of TreeWorld dot info forum, owner of Brisbane Palm and Tree Services at palmtreeservices dot com dot au). Also watch some of the not-in-english Scandanavian felling videos for an interesting view of some wise but non-American techniques.
5. Take good care of your equipment, and buy good equipment.
6. Use wedges to keep the kerf open when blocking up firewood. Getting your saw stuck and then yanking on it...not smart, or good for your saw.
7. Never cut with just one saw. Have a back-up to cut yourself out with, or to finish up if your main saw dies. Particularly important when felling - you never want to leave a half-felled tree standing while you go back home or to the shop to get your main saw fixed. Your 2nd saw doesn't have to be great, new, or even super powerful. It just needs to be dead reliable.
8. When in doubt, run a shorter bar. It will make your saw more lively, will mean fewer cutters to sharpen, will reduce the liklihood of sticking the tip into bad things - dirt (dull saw), other logs (kickback+injury) - and will encourage you to use the cut-and-roll technique mentioned in an earlier post, which is particularly good for firewood cutting.
9. Buy a cant hook or a timberjack. A cant hook + a log = a timberjack, though a store-bought timberjack can be nice sometimes.
10. When felling, use wedges and a sturdy ax/sledge. Have 2x or 3x as many wedges as you need before you start using them...same idea as (7), above.
11. If you are going to be felling stuff and want to use ropes to guide/control/encourage the fall, use REAL rope. Not Home Depot rope. I have been pleased with my collection of odd-length eBay pieces of Samson StableBraid, in appropriate sizes for the task at hand.
12. Practice. In a safe environment, try different techniques for cutting logs. Set up a test log at home, practice different bucking techniques.
12a. I have found that one of the most educational tools for new cutters is a "slow" saw and some 3-6" limbs. The new saw operator can actually watch the saw cut, watch the kerf widen/contract, etc. Things happen a lot faster with powerful saws and bigger logs, so this is a good way to get an introduction to the physics of wood, saws, and hinges. A little 30cc Echo and the remains of something "wet" and "soft" like a live box elder tree are ideal for this kind of learning experience.
13. Try to find a good saw shop. More than just an OPE shop, this is the place that focuses on saws, on having arborists/line clearance guys/loggers as their clients, and doesn't consider saws an incidental addition to the mowers, blowers, or tractor inventory. They'll have the parts/accessories and equipment that you actually need as a saw operator, not just a saw owner.
14. Always use good gas and great oil. 90+ octane, as little ethanol as possible, quality synthetic oil, and don't have it sit more than a month or so before you run what remains through your car and refill your saw gas container with new stuff.