You should see a doctor.

In the context of someone catching a cold, it may not be considered natural English to advise, "go to hospital". Generally, for a non-emergency situation like a common cold, it's more common to say "You should see a doctor" or "You might want to visit a clinic".

Going to the hospital is usually suggested in more serious or emergency situations. In the UK, it's somewhat more natural to say "go to hospital" (without 'the') for severe cases, while in the US, it's more common to say "go to the hospital". Again, both contexts usually refer to emergencies or more severe health problems.