For most people, taking vitamin C does not prevent catching a cold. Taken regularly it may marginally reduce how long a cold lasts, but starting it only once you are ill has little effect. The benefit is small and not a substitute for rest and fluids.
Zinc, particularly as lozenges started early in a cold, may slightly shorten symptom duration in some studies, but results are mixed, the ideal form and dose are unclear, and it can cause side effects such as nausea or an unpleasant taste. Excess zinc over time can cause other problems.
If you want to try a supplement, keep within recommended amounts and check with a pharmacist, especially if you take other medicines or have health conditions. Supplements are not a cure, and persistent or severe symptoms still warrant medical advice.