Regardless of the medical system itself, the terms are fairly the same in English:
Hospital is a place for definitive treatment of illness or disease. It can be a short stay or a long term admission but the point is: your purpose in being there-you are not leaving until you are healed.
A clinic has a different purpose. It does not have facilities to keep you overnight. It is intended for you to be diagnosed and either send you to a hospital for further treatment, or to treat you and send you home. A clinic is meant to give you definitive care for those that don't need hospital facilities.
A doctor is an individual that will treat you. In some places like the states, you go to a private doctor for diagnoses and treatment.
In some public medical systems or medical insurance systems you would first go to a clinic or the hospital has a clinic in it.
Your question is actually about the colloquialisms of the medical system of the location.
British colloquialism is: "go to hospital" Technically it is not a grammatically correct sentence structure-hence colloquialism