Moja sestra kaže da je to dobra vest.

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

• sestra is feminine singular
• so the possessive moj changes to moja

This is the same kind of agreement you also see in dobra vest:

• dobra matches feminine singular vest

So Serbian is doing something English does not do much: words around the noun often change their form to match it.

Sestra is in the nominative case because it is the subject of the sentence — the person doing the action of kaže.

So:

• moja sestra = the subject
• kaže = the verb

If sister had a different role in the sentence, it might appear in a different case.

Kaže is the 3rd person singular present form, meaning says.

It matches moja sestra, which is:

• 3rd person
• singular

So the logic is:

• ja kažem = I say
• ti kažeš = you say
• on/ona kaže = he/she says

Here, moja sestra kaže literally works like my sister says.

Because Serbian usually does not need subject pronouns when the verb form already shows the person and number.

So instead of saying:

• Ona kaže...

you can simply say:

• Moja sestra kaže...

or even just:

• Kaže...

if the subject is already clear from context.

This is very normal in Serbian. English usually requires the subject pronoun, but Serbian often leaves it out.

Here da introduces a clause meaning that.

So:

• Moja sestra kaže = My sister says
• da... = that...
• da je to dobra vest = that it/that is good news

This is one of the most common uses of da in Serbian: linking a main clause to what someone says, thinks, knows, wants, etc.

Because je is a clitic. In Serbian, clitics usually go very early in the clause, often in the second position.

In this clause:

• da is first
• so je comes right after it

That gives:

• da je to dobra vest

not:

• da to je dobra vest

This word order can feel strange to English speakers, but it is very standard Serbian clitic placement.

Je is the 3rd person singular present form of biti (to be), meaning is.

So inside the clause:

• to = that / it
• je = is
• dobra vest = good news / a good piece of news

Even though English learners often first meet sam, si, je, smo, ste, su as forms of to be, in Serbian these short forms behave specially and often move into clitic position, which is why je appears where it does.

Because dobra must agree with vest, and vest is feminine singular.

So the adjective dobar changes like this:

• dobar = masculine singular
• dobra = feminine singular
• dobro = neuter singular

Since vest is feminine, you need:

• dobra vest

not:

• dobar vest

Because in Serbian, noun gender is not determined only by the ending.

It is true that many feminine nouns end in -a, but not all of them do. Vest is one of the feminine nouns that end in a consonant.

So you simply have to learn its gender as part of the word:

• vest = feminine

You can see that it is feminine from agreement:

• dobra vest
• not dobar vest

To here means something like that or it, referring to some information, event, or situation already known from context.

In Serbian, to je... is a very common structure:

• To je problem.
• To je istina.
• To je dobra vest.

So to is pointing to an idea or situation, not to a masculine or feminine noun specifically. That is why it appears in the neuter form to.

Because Serbian and English organize this idea differently.

In Serbian:

• vest = one item of news / one piece of news
• vesti = news items / pieces of news

In English, news looks plural but grammatically acts like a singular mass noun. Serbian does not work that way here. It uses a normal countable noun:

• dobra vest = literally something like a good piece of news

So the singular vest is completely normal.

Yes, you may also see vijest.

This is mainly a pronunciation/spelling standard difference:

• vest = common in Ekavian Serbian
• vijest = common in Ijekavian varieties and also in Croatian/Bosnian

So depending on the variety, you may encounter either form. In this sentence, vest is perfectly standard Serbian.

Because Serbian does not have articles like English a/an/the.

That means Serbian often leaves definiteness to be understood from context.

So:

• sestra can mean sister / the sister / a sister, depending on context
• vest can mean news / a piece of news / the news item, depending on context

If Serbian wants to be more specific, it can use words like:

• taj = that
• ovaj = this
• jedan = one / a certain

But in your sentence, no article-like word is needed.

A rough pronunciation is KAH-zheh.

Helpful letter notes:

• ž sounds like the s in measure
• j in Serbian usually sounds like English y in yes
• je is roughly ye

So parts of the sentence sound approximately like this:

• Moja ≈ MO-ya
• sestra ≈ SES-tra
• kaže ≈ KAH-zhe
• da je ≈ da ye
• dobra vest ≈ DO-bra vest

That is only an approximation, but it is a good starting point.