There are many different causes for sore throats. Some causes are easily treated, some require antibiotics, and some cannot be treated at all -- you just have to wait out the cause, although over-the-counter medicines can help relieve the pain.
Coughing by itself can make your throat feel sore. So can postnasal drip -- mucus from the nose dripping down the back of the throat.
If you have a dry cough, cough suppressants -- some of which you can buy without a prescription -- can quieten the cough and cut down on throat irritation. If you are coughing because of postnasal drip, I believe you should not use cough suppressants -- I have seen people get pneumonia that way, since if the suppressants work you may not be able to clear your lungs properly. Sometimes decongestants can help reduce the postnasal drip and thus reduce the cough, but with viral colds decongestants may not do very much. Humidity and drinking fluids will help, since they will keep the mucus from getting thick -- thick mucus is much harder to cough out than thin mucus.
Although there are other bacteria that can attack the throat, the most common bacterial cause of strep throat is Streptococcus pyogenes, or "group A strep". (As with other bacteria, there are several different kinds of strep, which are similar overall but differ in some biochemical details -- and, because of those details, in the way they infect people. S. pyogenes, or group A, is the kind that usually infects throats and causes scarlet fever.)
The classic strep throat is very red (the description in some textbooks is "beefy red" or the colour of raw beef). Often you will see white or green "exudates" or pus on the tonsils, which are also swollen. You may also see little dark-red, almost purple, spots on the back of the roof of the mouth and on the uvula (the little "punching bag" that hangs from the center of the back of the mouth). The tongue may be very red with little white spots, like a strawberry (the "strawberry tongue") or white with little red spots, like a strawberry dipped in white chocolate. Sometimes a person with strep throat will have very bad breath, but this is not always true.
Usually a person with strep throat will have swollen and tender lymph nodes at the front sides of the neck, and the throat will hurt so much that even swallowing liquids is painful. Coughing is rare with strep throat, unlike viral or dryness-related soreness. Temperature may go up to 103 or more.
There are other things that can happen with strep throat. Scarlet fever is the best-known: this is a combination of strep throat with a rash, usually on the lower abdomen and spreading from there to the trunk, that consists of fine red bumps with a sandpaper-like consistency. The more important complications of strep throat, which usually do not happen unless the infection is left untreated for a long time, include
Fortunately, group A streptococcal infections are very easy to treat. No one has ever reported a case of strep throat that was not cured by penicillin. Penicillin G or V (the "plain" penicillins) usually work just fine, but have to be given as a shot (penicillin G) or 4 times a day on an empty stomach (penicillin V). Although some infectious disease specialists frown on it, amoxicillin also works well on strep, and has two advantages over penicillin V: it can be given 2 or 3 times a day, and it doesn't matter much if you give it on a full or an empty stomach. (Actually, there's a third advantage: it tastes better than penicillin V. As a pediatrician, that is important: antibiotics don't work well if the patient won't take it.) Other antibiotics, such as erythromycin, will work almost (but not quite) as well as the penicillins and can be used in patients allergic to penicillin.
One final note: although it is usually not associated with strep throat, group A strep can also produce a "necrotizing fasciitis" (the infamous "flesh-eating bacteria"). There is still little known about how group A strep gets this out of hand. The treatment still involves penicillin, but in most cases surgical "debridement" (removing the dead tissue) must also be done to help keep the bug in check.