The Components
Since I ride road bicycles, I prefer drop handlebars. When seen from the side, they form an U lying on its side. They afford many different hand positions: on the flat top part, holding the brake handles, on the ends, or between the ends and the handles. With some, the curved part is nearly semicircular; I prefer the kind made by Modolo and others where the curved part has another straight section directly below the brakes. It's not so convenient to grasp a curved part of the handlebar. Some handlebars have one or two grooves that help installing brake and (for Campagnolo Ergopower) shifting cables. In comparison, straight handlebars like those usually used on mountainbikes only afford a single hand position, and a fairly unnatural one with the palms facing down that makes my hands hurt after a while.
The handlebars are held in the center by the stem. On road bicycles, the stem is angled downward to compensate for the angle of the steering tube of the bicycle frame, such that the top of the stem that holds the handlebars is horizontal. There are stems that angle upward, but this raises the handlebars and forces the rider into a more upward position that increases wind resistance.
There are various types of aero bars, also called triathlon bars. The most common types are clamped to the top of the handlebars. There are many different types. My current favorite is Syntace, despite the prices that border on extortion (little required pieces of plastic need to be bought separately at unreal prices). They clamp on the thick inner part of the handlebars, leaving more room for holding the top of the handlebars, they have a very low height and comfortable and adjustable armrests, and they have just the right length and a steep front that doesn't require bending the wrists too much.
Are aero bars necessary? This depends on the kind of riding you do. I began using them when I had a 19-km ride to work every day into a persistent headwind. They also offer an edge when it's my turn at the front during group rides. They are not useful for short trips, and they can be dangerous in the city because it takes longer to reach the brakes, and because they move the center of gravity forward and increase the danger of the rear wheel losing contact with the ground when braking. They cannot be used when climbing hills. They should never be used with old-fashioned brake levers with brake cables coming out of the top because it's easy to get one's gloves caught when reaching for the brakes.
Adjusting the Handlebars
The lower part of the stem should be long enough to raise the top to about the level of the top of the saddle, or a few centimeters lower. Lower position decrease wind resistance but put more strain on the neck muscles. The top of the stem is not normally set higher than the top of the saddle. The length of the top part of the stem should be chosen depending on the length of the top tube of the frame; usually it's longer for people with a long upper body compared to the length of the legs. Care should be taken to choose a solid stem that does not flex when pulling hard on the handlebars, like when standing to get up a steep hill.
Some people recommend to adjust the drop handlebars such that their ends point to the center of the seat tube of the frame (the one that runs from the bottom bracket to the seat post). This makes me feel like I am sliding off the ends, so I adjust mine to be almost horizontal, angled downward very slightly. The straight section in the curve allows an angled hand position anyway. This angle must be chosen before anything else is installed.
The brake levers are adjusted next after choosing the angle of the handlebars. They should be installed such that they can be comfortably held with your hands on the brake lever body, and they can be reached easily when the hands hold the curved part of the handlebars. For me, this means that the rubber body is about horizontal, installed at the forward-most (vertical) section of the handlebars. This must be done before installing the handlebar tape and cutting the cables; it's very difficult to make adjustments later. Do not use "safety brake levers" that have an extra lever that extends back towards the stem; they brake poorly and give a false sense of security.
The aero bar angle is also pretty much a matter of taste. I prefer the aero bars angling slightly up towards the front end. Any larger angle would hurt my forearms resting on the armrests.
Handlebar Tape
The handlebar, and many aero bars, need to be wrapped with tape. The tape should have an adhesive strip running along the back, and should be a soft foam or cork material. I have found Cinelli to be the best; unfortunately it's expensive. Real cork does not last long. The cheap material that looks like artificial leather is uncomfortable. Choose a dark or mottled color that won't look dirty quickly.
Before wrapping the tape, install any brake and shifter cables that run to the brake handles or the bar-end shifters, using the grooves in the handlebars if available. Tape the cables to the handlebars with electrical tape to they don't come off all the time during wrapping.
Wrapping begins by attaching the short strips to the metal strap that holds the brake levers to the handlebar. Fold back the rubber coating of the brake lever body. Next, cut off one edge of one of the long handlebar strips so it tapers towards the end, remove the end of the backing to expose the adhesive strip, and begin wrapping at the end of the handlebar (not at the center near the stem). Wrap clockwise on the right side and counter-clockwise on the left side, seen from the rear of the bicycle (this prevents unwrapping when holding the handlebars on the top later). Wrap under tension, but not enough to stretch the tape significantly. The loops should be spaced such that the adhesive strip is just barely on the handlebar, not on the previous loop of the tape. I tend to space the loops more closely near the end and wider near the stem.
When you reach the brake lever, make sure that the last loop folds against the brake lever body. The next loop extends all the way to the other side of the brake lever body. Some people recommend to then fold the tape to run back along the lever to the lower end and then back over the previous loop than spanned the body, only in the other direction so it crosses the previous loop, but I find it makes the brake lever section too fat and wastes too much tape. Then complete wrapping the tape to the place where the handlebar gets wider near the stem, taper the ends again, and seal the end with electrical tape or the adhesive strips in the handlebar package. I usually have to re-wrap the last part after the brake lever a few times to get the length just right.
If you have aero bars that clamp on the narrow part of the handlebars, not the wide part near the stem, remember to leave enough of the handlebars untaped to install the aero bar brackets. When folding back the rubber brake body coat no part of the handlebar metal should be visible. For this purpose, put a 10cm piece of tape across the clamp that holds the brake handle, with the ends disappearing under the rubber cover of the handlebars after they are peeled back in place.
See also an article on "Headset"