Drop handlebars really need to be taped or otherwise covered. This is not just about cushioning/insulation, but also about traction. If your handlebars are slippery, they'll force you to hold them with a "death grip" to keep from sliding, which will lead to hand/wrist pain.
Before you start, make sure you have all of the necessary tools and materials at hand. Wash and dry your hands carefully, not just to keep the tape from getting soiled, but also so that oils from your skin won't compromise the connection between the tape and the handlebars.
You will need a good pair of scissors, make sure they're where you can reach them without having to let go of the tape. Once you start to wind the tape you must never release the tension on it until the job is complete.
If the phone rings while you're in the middle of the job, let the caller leave a message.
Roll the rubber hoods forward to expose the bodies of the levers. The tape always goes underneath the rubber hoods.
Generally, handlebar tape should be wound on starting at the ends of the bars and winding toward the middle. The reason for this is the direction of the overlap, specifically in the area of the "corners" (the upper, forward bend of the handlebar.) If you start at the ends of the bar, the layers overlap one another like the shingles of a roof, with the upper edge of the tape protected by the next winding up from it.
If you start from the middle and work toward the ends, the upper edges of the tape are exposed, and hand pressure will tend to roll them down and away from the handlebar.
The exception to this is thin plastic tape, which is usually immune to this sort of failure, if installed with sufficient tension.
Whether you start from the middle or from the ends, make sure to go in the same direction on both sides. I generally start from the underside of the bar, with the tape feeding outwards. You can do it the other way too, but make sure you follow the same pattern on both sides.
It is important to secure the cables to the bar snugly so that they don't move when the brakes are applied. I usually do this with electrical tape.
Some folks cut multiple short strips of electrical tape for this purpose, but I prefer to just wind a continuous open spiral of tape along the bar. This saves time and is more secure.
I usually use traditional cloth tape for this purporse, a strip running along the top/rear of the bar around the corners. This enhances the tape's traction, since, and adds a little bit of extra padding where it can do the most good as well.
Good tension keeps the tape from slipping, and also reduces the tendency of the tape to wrinkle where it goes around the bent parts of the handlebar.
Modern tape is fairly thick, so you should try to keep the overlap pretty much constant, to avoid lumpiness.
I generally use Cinelli "Cork" tape, and these instructions are somewhat specific to this brand, but generally applicable to other similar tapes.
I use a pair of scissors to make a long diagonal cut so that the blunt starting end of the tape is only about half the full width of the tape. The diagonal cut is about 3 inches long, and the cut section matches the end of the handlebar.
I start with the square end at the bottom of the bar, pulling the loose end up and out.
The diagonal cut permits a constant overlap of the tape, so there's no thick lumpy section at the end of the bar.
As with the beginning, I do a diagonal cut for the end of the tape job as well, but the end section is a full width diagonal cut, about 6 inches long tapering to a sharp point. The adhesive finishing strip will then fit without wrinkling, since the tape thickness is constant right up to the cut end.
Bias-woven cloth tape was the norm from the late '70s well into the '80s. It is particularly associated with French bicycles, and the major brand, Tressoplast, is French.
This is also adhesive-backed, and is applied pretty much the same as padded plastic. It isn't all that stretchy, so you may need to apply a LOT of tension to keep it from wrinklin where it runs around bends in the handlebars.
Thin, non-adhesive plastic tape such as Benotto or Hunt-Wilde is best applied from the center of the bar towards the ends. The starting end is covered by the overlap near the stem, and the finishing end is tucked into the ends of the handlebars and secured by the plugs.
Foam grips, such as "Grab-on" brand were quite popular in the late '70s and early '80s. They can be a good choice for riders with long fingers, and they provide excellent insulation for winter riding. Some riders like to cover them with a layer of cloth tape to make the foam a bit firmer, and to protect it.
Installing foam grips is often quite a struggle. My favorite approach is to use hair spary a lubricant. Spray into the grips, while holding one end covered, and also spray copiously on the handlebar just before sliding the grip on.
I also keep a hypodermic syringe handy, filled with rubbing alcohol. If the grips get stuck part way on, I jab the needle through the foam at a shallow angle so that it will pierce through to the handlebar, and give a good squirt of alcohol. That will free the grip. This is also a good way to remove the grips when necessary without damaging them.
Installing foam grips with cable housing running under, as for "æro" brake levers, is especially challenging. If you must do this, get the housing segments carefully sized first, then thread a cable through the housing, and stick the end of the bare cable throught the foam grip before sliding it onto the handlebar. Once the grip is in place, pull on the bare cable to draw the housing through the grip.
Sometimes, foam grips may be counter-productive, due to the "two bump problem."