Hills
When climbing at less than a sprint, where you are not pulling on the handlebars, use the tops or the drops, staying away from the aero bars and the hoods. On a long plod, where there is nothing left from your approach momentum, feel free to sit up in the wind using the tops, as it won't noticably impact your speed -- nearly all effort is fighting gravity, not fighting wind.
Sprinting
If you start pulling on the bars for more power, go to the drops. If you're sprinting up a hill, the hoods are great. Stay away from the top bend and the back of the drops, and to an extent the tops when pulling on your bars. The importance of an effortless grip aside, your arms work most powerfully and efficiently when extended.
Cruising
If you're riding briskly, or against a headwind, stick to the aero bars or drops. If riding hard, put your head down and really jamb you hands into the curved section. This will get you out of the wind and balance your weight.
A strong tailwind, or a lack of any particular rush, makes the best position change to the ends of the drops, the hoods, or to the bend behind the hoods. These positions are most comfortable for long distances, or rides of no more than moderate intensity.
Avoid the flat section of the handlebars when cruising, but not totally. On a very long ride, I have a way of sticking to the drops until I wear out my arms and am forced to use the flat section, having "used up" all of the lower positions. Had I rested my arms before they tired by switching to the flat section -- or even to the hoods -- occasionally during the outbound ride, I would have had a larger total fraction of the overall ride spent in the more efficient tuck.
In general, however, you'll only want to use the flat section if you are riding up a hill in no particular hurry, are travelling very slowly, or are trying to give the other positions a rest.
Burping
To burp, go to A and tilt your head back. Don't even bother trying to get the air you swallowed out when on the drops. Not only is there no rush, but all you'll do is give yourself heartburn without accomplishing anything. You usually don't need to sit up to burp.
Incidently, swallowing air in an attempt to make burping easier does not work. You will swallow more air than you subsequently expel, making the problem worse. If you tend to swallow air when drinking from a water bottles, try sucking on the nipple and filling up your whole mouth, rather than squirting a bit in your mouth, and swallowing the water with a mouthful of air, over and over again. If you drink anything carbonated, the best bet is to get a half-dozen solid burps in before you get back on your bike.
