"You should go see a doctor."
Yeah, and 看病 is only for illness, not suitable when one was just injured in an accident.
看醫生／看医生 is the most common phrase to use in the conversation.
看病 is not often used as one could literally interpret it as "look disease"..
Including 去 connotes a sense of distance. It is not wrong without 去 for the same situation.
”你应该去看一个医生“ Was marked wrong 2019-02-23. I think you're allowed to say “一个” to indicate "a" but it would be acceptable without it.