Between "... go to the doctor" and "... see the doctor", "see the doctor" is better because we go to places but we see people, eg: "You're still coughing!?!? Did you see the doctor?"
Between "... a doctor" and "...the doctor" both are grammatically correct but have different meanings.
The definitive article "The" is used for a specific noun eg: "The doctor that performed my surgery is the best in the country"
The indefinite article "a" is used for singular noun that is nonspecific eggs "Pick a flower, any flower"