My dictionary says both functions like a conjunction that in English.
They could both be translated as 'that' in some circumstances, but I'm not sure that's very useful information, as 'that' is a word with many functions and meanings in English.
It might be more useful to remember their function, which is to report speech.
English has the concept of indirect and direct reported speech, and so does Korean.
Indirect reported speech is more complicated than direct reported speech in Korean as there are four endings commonly used.
(ㄴ)다고 is used for indirect reporting of declarative statements (straightforward facts):
이 약을 하루에 두 번 먹어야 한다고 말했어요 => She said that I should take the medicine twice daily.
(느)냐고 is used for indirect reporting of interrogatives (questions):
어디 가냐고 했어요 => He asked where I was going.
자고 is used for indirect reporting of propositives (suggestions):
사장님께서 태국에 사무실을 열자고 말씀 했습니다 => the boss suggested we open an office in Thailand.
라고 is used for indirect reporting of imperatives (instructions):
춤을 추라고 했어요 => we were told to dance.
라고 is also used for indirect reporting of structures with the copula, 이다:
나쁜 놈 이라고 했어요 => they said he was a bad man.
Direct reported speech is easier - it always uses (이)라고:
"사랑해요" 라고 했어요 => She said "I love you".
개는 "멍멍" 이라고 했다 => the dog went "woof woof".
"뭐라고" could be seen as a special case. You can see the similarity between 뭐라고 and the other forms using 라고 above, but it might be simplest to think of "뭐라고" as a word you use when asking what someone said - and 의사가 뭐라고 말했어요? is an example of this.