Learn Hardcore Croatian: Doktorica kaže da moj trbuh izgleda zdravo, ali da trebam više vježbati. - The female doctor says that my stomach looks healthy, but that I need to exercise more.

Because the sentence is talking about a female doctor, it uses doktorica.
If you were talking in general about doctors (without specifying gender), people often still say doktor in the masculine as the default, but when the person is clearly female, doktorica is the usual form in everyday speech.

What does da mean here, and why is it repeated: da moj trbuh izgleda zdravo, ali da trebam više vježbati?

Here da is a conjunction meaning “that” in English, introducing a clause that reports what someone says or thinks.

The first da introduces the first clause:

da moj trbuh izgleda zdravo = that my stomach looks healthy

The second da introduces the second, contrasting clause:

ali da trebam više vježbati = but that I need to exercise more

Repeating da after ali is quite natural and common in Croatian, especially when the second part is a full clause with a different verb (trebam vs. izgleda).
You could also hear a shorter version:

… da moj trbuh izgleda zdravo, ali trebam više vježbati.

Both are correct; the repeated da just makes the structure a bit clearer and more explicit.

Why is it izgleda zdravo and not just je zdravo?

izgleda zdravo literally = looks healthy
je zdravo = is healthy

So the difference is the same nuance as in English:

My stomach looks healthy (judging by appearance, examination)
My stomach is healthy (stating a fact about its condition)

In medical or observational contexts, izgleda zdravo is very natural, because the doctor is describing what they can see or infer.
je zdravo is also possible but slightly more absolute; it sounds more like a firm statement of fact.

Why does zdravo end in -o? I thought “healthy” was zdrav or zdrava.

The base adjective is:

masculine: zdrav
feminine: zdrava
neuter: zdravo

Here zdravo is used after the verb izgleda. In Croatian you will commonly see both:

izgleda zdrav
izgleda zdravo

Both are used in real language. You can think of zdravo here in two ways:

As the neuter singular form of the adjective, used in a more “neutral” way.
Functionally a bit like an adverb modifying izgleda (looks healthy).

You do not need to change zdravo to agree with trbuh; izgleda zdravo is a fixed, natural pattern.

What tense and aspect is kaže, and how does it compare to govori?

kaže is the 3rd person singular present tense of kazati (to say).
govori is the 3rd person singular present of govoriti (to speak, to talk).

In this sentence:

Doktorica kaže… = The doctor says… / The doctor is saying…

kazati is more about a specific act of saying something (a concrete message).
govoriti is more about speaking/talking in general or talking for some time.

So:

Doktorica kaže da… = She states/tells that… (very natural here)
Doktorica govori da… = She says that… (possible, but feels more like she (generally) says/claims that…)

For reporting exactly what the doctor told you, kaže (or rekao je / rekla je in the past) is the most natural choice.

Why is it trebam and not treba or moram?

There are three related ideas here:

trebam – I need (to)
- trebam is the 1st person singular: I need.
- It’s a personal verb form; the subject is clearly “I”.
treba – can be:
- 3rd person singular: he/she/it needs
- or an impersonal form: one needs / it is necessary
  Example impersonal use: Treba više vježbati. = It is necessary to exercise more.
moram – I must / I have to
- Stronger obligation than trebam.

In your sentence:

trebam više vježbati = I need to exercise more. (personal, about me)
moram više vježbati = I must / I have to exercise more. (sounds stronger, more obligatory)

The doctor is saying you need more exercise; trebam fits that nuance well.

Why is vježbati in the infinitive form, and why isn’t there a word like “to” before it?

In Croatian, after verbs like trebati, morati, htjeti, etc., the next verb is normally in the infinitive, without any extra word:

trebam vježbati = I need to exercise
moram učiti = I have to study
želim spavati = I want to sleep

English uses “to” as an infinitive marker: to exercise, to study, to sleep.
Croatian does not; the infinitive is simply the base form of the verb, here vježbati.

So:

trebam više vježbati literally = I need more exercise / I need to exercise more
but structurally just: need + more + exercise-INF.

Could I say trebam više vježbe instead of trebam više vježbati?

You can, but the meaning is slightly different:

trebam više vježbati = I need to exercise more (focus on the activity, the action of exercising)
trebam više vježbe = I need more exercise / more practice (more like I need a greater amount of exercise/practice)

Both are correct Croatian.
In the context of a doctor advising you about your health, trebam više vježbati is more direct and natural, because it tells you what you should do (exercise more), not just what you need more of.

Is the comma before ali obligatory in Croatian?

Yes. In standard Croatian, you must put a comma before ali when it joins two clauses:

… izgleda zdravo, ali da trebam više vježbati.

This is different from English, where the comma before but is sometimes optional in short sentences.
In Croatian, the rule is strict: ali (but), nego (but rather), no (but), etc., normally require a comma when they link clauses.

Can I change the word order to da izgleda moj trbuh zdravo?

That word order is not natural.

The usual, natural order is:

da moj trbuh izgleda zdravo

In Croatian, possessive adjectives like moj usually come right before the noun they modify (trbuh). Splitting them (izgleda moj trbuh) sounds awkward unless there is a strong emphasis or some special context, which there isn’t here.

So you should keep:

moj trbuh together
then the verb izgleda
then zdravo.

What’s the difference between trbuh, stomak, and želudac?

All three are related to the idea of the stomach/abdomen, but they’re used differently:

trbuh
- Literally: belly / abdomen.
- Very common and neutral in standard Croatian.
- Used both in everyday and medical contexts for the general belly area.
stomak
- Also belly / stomach.
- More typical in some regional varieties and in Serbian/Bosnian/Montenegrin.
- In Croatia, trbuh is more standard, though people will understand stomak.
želudac
- Specifically the stomach organ (inside your body).
- Used when talking about digestion, ulcers, etc.:
  - Boli me želudac. = My stomach (organ) hurts.

In your sentence, trbuh makes sense because the doctor is probably talking about your belly/abdominal area looking healthy, not specifically the inner organ.