Ja učim hrvatski, a moja sestra piše poruku.

Yes. Croatian often drops subject pronouns because the verb ending already shows who is doing the action.

• učim = I learn / I am learning
• piše = he/she writes / is writing

So Ja učim hrvatski can also simply be Učim hrvatski.

Here, ja is included for emphasis, clarity, or just style. It can sound a bit more explicit, especially because the sentence contrasts I with my sister.

A often translates as and, but it usually has a slight contrastive feel. In this sentence it is something like:

• I am learning Croatian, while my sister is writing a message.
• I am learning Croatian, and my sister is writing a message.

So a connects two ideas and gently contrasts them: I am doing one thing, my sister is doing another.

It is not exactly the same as i, which is the more neutral word for and.

• i = simple addition
• a = addition with contrast or comparison

Because a is joining two independent clauses:

• Ja učim hrvatski
• moja sestra piše poruku

Each part has its own subject and verb, so a comma is normally used before a in Croatian.

Učim is the 1st person singular present tense of učiti (to learn, sometimes to study).

So:

• ja učim = I learn / I am learning

Some related forms are:

• ti učiš = you learn
• on/ona/ono uči = he/she/it learns
• mi učimo = we learn
• vi učite = you learn plural/formal
• oni/one/ona uče = they learn

Yes, hrvatski is originally an adjective meaning Croatian, but in sentences like this it is commonly used on its own to mean the Croatian language.

So:

• učim hrvatski = I am learning Croatian
• literally, something like I am learning Croatian [language]

This is very common in Croatian with language names.

Because it is the direct object of učim, so it is in the accusative case. But for this kind of masculine inanimate form, the accusative singular looks the same as the nominative singular.

So:

• nominative: hrvatski
• accusative: hrvatski

That is why you do not see a visible change here, even though grammatically it is the object.

Because moja has to agree with sestra in gender, number, and case.

• sestra is feminine singular
• so the possessive must also be feminine singular
• therefore: moja sestra = my sister

Compare:

• moj brat = my brother
• moja sestra = my sister
• moje dijete = my child

Piše is the 3rd person singular present tense of pisati (to write).

So:

• moja sestra piše = my sister writes / is writing

Croatian present tense can often mean either:

• a general action: writes
• an action happening now: is writing

The context tells you which meaning is intended.

Because poruku is the accusative singular form of poruka (message), and it is the direct object of piše.

• nominative: poruka = a message
• accusative: poruku = a message as the thing being written

So:

• poruka je duga = the message is long
• piše poruku = she is writing a message

This is a very common feminine noun pattern:

• nominative singular in -a
• accusative singular in -u

It can mean both.

Croatian present tense often covers both:

• I learn Croatian
• I am learning Croatian

Likewise:

• moja sestra piše poruku can mean my sister writes a message or my sister is writing a message

English separates these two ideas more clearly than Croatian does. In Croatian, context usually makes it clear.

Yes. Croatian word order is more flexible than English because case endings help show what each word is doing.

The neutral version here is:

• Ja učim hrvatski, a moja sestra piše poruku.

But other orders are possible for emphasis, for example:

• Hrvatski učim ja, a poruku piše moja sestra.
• Moja sestra piše poruku, a ja učim hrvatski.

These alternatives can sound more marked or emphatic. For a learner, the original word order is the safest and most natural to start with.