When it comes to enjoying a fine cigar, there is a right way (the Luxe way) and a wrong way. Like any worthwhile past time, it involves a series of rituals which serve to prolong one's enjoyment. It begins with the selection of the cigar.
When selecting a cigar, always look for "hand made", a statement which generally ensures that you're purchasing a cigar made of long-filler (not the scraps of tobacco left over from premium cigar production).
Next, when selecting your cigar from amongst the many others in the box, don't be bashful. Inspect as many cigars as needed to find one free of blemishes; these could be small holes in the wrapper, cracks, a particularly large vein, or an inconsistent structure.
To check for the latter, gently squeeze the cigar between the face of your thumb and your index & middle fingers, moving along its length. A well rolled cigar will result in either a dense, spongy structure running the entire length OR a firm structure feeling almost light in weight. If you find a spongy cigar with some hard spots, do not purchase it; it'll most likely have draw issues & burn unevenly. Inspect another.

Also, smell the foot and lower third of each cigar you consider purchasing...this should give you a hint of the smoke's character, whether it be spicy, etc. Pick one that has a personally pleasing aroma.
Once you've selected your cigar, its time to cut it. The head, or wrapped end of the cigar, is from where one draws smoke. (The foot is the open end of the cigar, where you can see the filler tobacco.) When making a straight cut, also called a guillotine cut, snip off only the cap of the cigar (use the visible line of the cap as a guide); imagine snipping just the tip of your index finger where it begins to taper or curve.

In the same way, if you select a cigar with a pointed end (a figurado) clip off just enough so that your cut exposes an area of the inner filler with a slightly smaller diameter than the body of the cigar.
(In Europe, the Cat's Eye or V-cut is popular, utilizing a v-shaped blade to cut a recessed V into the head of the cigar. This has the advantage of a cooler smoke with no annoying bits of tobacco getting into one's mouth; nonetheless it allows a build-up of tar & nicotine at the interface while sometimes restricting the draw of the cigar. Same goes for the Punch cut, which pierces a small hole into the cap of the cigar.)
Now you get your first taste of the cigar. After gently blowing/ brushing away any loose tobacco from the freshly cut end, take a few gentle pulls of the cigar, pressing the tip of your tongue lightly against the now exposed end of the cigar. This gives you a pre-light impression that serves as an introduction. Time to light-up.

The proper way to light a cigar is NOT debatable. NEVER touch the flame of your heat source (match, butane lighter, or cedar stick) to the foot of the cigar; it will impart the flavors of your fuel to the cigar. Also, in the case of butane, you would char the foot of the cigar, causing a change in the true flavor of the stogie. When using a match, allow it to burn a few seconds before using it to light the cigar...this allows the sulfur to subside, saving you from notes of rotten eggs in your smoke.
When lighting, hold the foot of your cigar an inch or more above the tip of the flame...depending on how large your flame is. Holding the cigar at a 45 degree angle, rotate the cigar so that the foot can be heated slowly. After some time (patience, patience), you will begin to see the outer edge of the wrapper near the foot become lighter in color. Once you see this light ring the entire way around the foot, you're ready to puff (DON'T INHALE! EVER!).

Maintaining the 45 degree angle and 1" distance from the flame, begin to gently puff (NOT PULL!) on the cigar while continuing to rotate. If properly toasted, your cigar should catch fire within seconds. Don't be alarmed at the flaming cigar...continue to puff and hold the cigar in the proper position until you make 2 complete turns while puffing. This should ensure that you are evenly lit.
Inspect the foot of the cigar by blowing on it, making sure the entire surface glows when lit. If not, repeat the step of puffing and rotating.
Now, after lighting, allow your cigar to cool a bit; don't puff on it for 30 seconds to a minute. When smoking, puffing too frequently will cause the cigar to overheat and it may crack along its wrapper and unravel.
Now you're smokin'! Take your first post-ignition puffs. Allow the smoke to rest on your palate briefly while you contemplate the flavors, aromas, and strength of the smoke. Exhale. If you've made a good choice, you'll be rewarded with a delicious palate and inviting aroma.
When smoking, NEVER leave a cigar sitting in your mouth...despite how cool it may look, a saliva soaked cigar is never good for many reasons. Take a puff. Remove. Repeat.
If you're a new enthusiast, don't be afraid to ask your tobacconist for suggestions. Smoke on a full stomach with water, cleaning your palate after each puff while slowing your body's absorption of nicotine.
The ritual of selecting, cutting, and lighting your smoke should be done without haste. It is a part of the experience of each cigar and should be savoured just as one appreciates a smoke's flavors and aroma. Enjoying a cigar is an unhurried, restful past time...enhanced by great company, conversation, and ambiance. Indulge.