Whether you need a cough suppressant or an expectorant depends on the type of cough you have. Coughs that produce mucous or phlegm are usually helping your body's immune system fight an infection and should be encouraged with an expectorant. Dry coughs that do nothing but keep you awake at night don't often benefit from an expectorant. For those coughs, you can try a cough suppressant.
There are lots of cough suppressants on the market. Most of the cough syrups sold over the counter haven't been shown to work. Honey works a little, but the urge to cough is such a complicated process that researchers are still trying to find a medicine that will truly suppress coughs.
Some coughs are caused by mucous draining from the nose into the back of the throat. Those coughs will often benefit from the use of an antihistamine with a decongestant. It doesn't do anything about your cough, but it stops the mucous from dripping back there and tickling your throat -- which should give you some relief.
The problem with antihistamines and decongestants is that they can cause anxiety. Since nasal drip is a nighttime problem, taking the drugs to help you sleep could backfire, stopping the drip but keeping you up anyway.
For really irritating coughs, or coughs that last more than three days, your doctor may be able to prescribe a stronger cough suppressant. More importantly, seeing the doctor for a persistent cough could identify the underlying cause, which is the only real way to fix a cough.