As with most viruses lacking specific antiviral drugs the best thing you can do for a cold is to treat the symptoms, rest, and hope that it is a rhinovirus and not an adenovirus.
To hasten convalescence:
To treat the symptoms:
-soup + painkiller for throat pain
-cough medicine if there is a cough
Zinc ions are important to the catalytic site of a number of enzymes and do play a role in the immune system. Thus it is important to make sure you are not zinc deficient (nor deficient in any other mineral related to immunity). However, taking more than the basal amount of zinc isn't really helpful (the excess will just be excreted). Having it on hand will not cause your body to generate more immune proteins, and truly excessive amounts can actually be detrimental.
Basically this is something that your body will have to heal itself. So step one: if you normally do not get enough of any vital mineral in your diet make sure you do. Eat a balanced diet and don't be afraid to eat a little more than you normally might, you need the minerals, metabolites, and resultant calories. Step two: rest. Get enough sleep and don't exert yourself if you can avoid it. You need to let your immune system work.
Feel better alx!