When you have small health problems like colds, flus, and infections, you usually go to a doctor's office or a clinic. You can call this "going to a doctor" or "going to the doctor".
This is different from "going to a hospital". A hospital is somewhere that you go when you have major health issues that require you to stay several days. You'd go to a hospital if you're getting surgery, if you had a heart attack, or if you're giving birth.