In the US, a Family Doctor or General Practitioner should be able to take care of these problems. But the specialists who treat the very severe forms of the flu are the Infectious Disease specialists. There are a handful of doctors who are called Virologists who work specifically with viruses, many concentrate on treatment of HIV/AIDS.
There is nothing for doctors to do about a common cold - it is a virus that will work itself out. You can take over-the-counter remedies/medications that help with cough, stuffiness, sore throat, etc., but there is no cure for the common cold. Ask your pharmacist to recommend symptomatic treatments for your specific illness.
Flu on the other hand, can be deadly - especially in those who are very young, very old, or with compromised immune systems, but even then the doctors who will be involved most often in the care and treatment are Family Doctors, General Practitioners, Pediatricians and Internal Medicine specialists.