התסמין הראשון של המחלה היה עייפות, ואחר כך באו גם כאב ראש ושיעול.

More or less, yes. Both mean symptom.

• תסמין is the standard Hebrew word and can sound a bit more formal or medical.
• סימפטום is a very common loanword from European languages and is also widely understood.

So in this sentence, התסמין הראשון is completely natural and standard.

Because in Hebrew, when a noun is definite, an adjective describing it is usually definite too.

So:

• תסמין ראשון = a first symptom
• התסמין הראשון = the first symptom

This is a very important pattern in Hebrew:

• הילד הקטן = the small boy
• ספר מעניין = an interesting book
• הספר המעניין = the interesting book

של means of.

So:

• של המחלה = of the disease

The word המחלה has ה־ because it refers to a specific disease, not just any disease. Hebrew keeps the definiteness on the second noun after של:

• ספר של מורה = a book of a teacher
• ספר של המורה = a book of the teacher

So התסמין הראשון של המחלה is literally the first symptom of the disease.

Because the sentence is in the past.

In Hebrew, present-tense sentences like X is Y often do not need a verb:

• הוא עייף = he is tired

But in the past, you normally do need a form of היה:

• הוא היה עייף = he was tired

So here:

• התסמין הראשון של המחלה היה עייפות = the first symptom of the disease was fatigue

Because the verb agrees with the grammatical subject, not with the word after it.

The subject is:

• התסמין הראשון = the first symptom

And תסמין is masculine singular, so the verb is masculine singular:

• היה

Even though עייפות is feminine, it is the predicate here, not the subject.

A useful comparison:

• העייפות הייתה התסמין הראשון = the fatigue was the first symptom
  Here עייפות is the subject, so the verb becomes הייתה.

Here it is singular.

Even though ־ות is often a feminine plural ending, some Hebrew nouns ending in ־ות are singular abstract nouns. עייפות means fatigue or tiredness as a general condition.

So in this sentence, עייפות is one symptom, not multiple fatigues.

This is something English speakers often need to get used to: form and meaning do not always match in a simple way.

Yes, literally באו means came, but Hebrew often uses come in a broader sense, just like English sometimes does in expressions such as the pain came back or the fever came on.

Here באו גם כאב ראש ושיעול means something like:

• a headache and a cough also appeared
• then a headache and a cough came on too

So באו is natural and vivid. A more explicitly medical-style verb could be הופיעו = appeared.

Because the subject is plural in meaning:

• כאב ראש ושיעול = headache and cough

That is a pair of symptoms, so the verb is plural:

• באו = came for they came

Even though כאב ראש is one expression meaning headache, together with שיעול it becomes a compound subject of two things.

Hebrew often puts the verb first when introducing new events or new items, especially with verbs like came, appeared, arrived, and similar verbs.

So:

• ואחר כך באו גם כאב ראש ושיעול

sounds very natural for and after that, a headache and a cough also appeared

If you moved the subject first, it would sound more marked or would shift the emphasis.

כאב ראש is a fixed Hebrew expression meaning headache. Literally, it is pain of head or head pain.

It is written as two words because it is built like a noun phrase, not a single fused word.

There is no ה־ here because it is indefinite:

• כאב ראש = a headache
• כאב הראש = the headache

The same idea applies to שיעול here: it is also indefinite. The sentence is introducing additional symptoms, so indefinite forms are natural.

They help organize the sequence of symptoms.

• אחר כך = after that / afterward / later
• גם = also / too

So the sentence first gives one symptom, and then adds more:

• first: עייפות
• later: גם כאב ראש ושיעול

This makes the progression very clear: The first symptom was fatigue, and afterward a headache and cough also appeared.