I had a cough, now my throat feels uncomfortable. When do I know I should see a doctor?

Post-nasal drip? This could be due to post-nasal drip which results in cough associated w/throat irritation. If your cough is lasting for more than 3 weeks or if short of breath, you should see your doctor for diagnosis and treatment. There are prescription nasal sprays that are very effective. And, if your cough is due to some other diagnosis, your doctor can treat you once the precise etiology is known.
7-10 days. If your symptoms don't resolve in 7-10 days then I would see a doctor. Within that time it is probably a viral infection (unless high fever). If it persists beyond 7-10 days, it may be a sinus infection.