Moj sin traži posao.

Because moj has to agree with sin.

• sin = son
• sin is masculine singular
• so the possessive adjective must also be masculine singular

That gives:

• moj sin = my son
• moja ćerka = my daughter
• moje dete = my child

So moj is the correct form for a masculine singular noun like sin.

Serbian normally does not use articles.

So where English says:

• my son is looking for a job
• my son is looking for the job

Serbian can simply say:

• Moj sin traži posao.

The exact English article often has to be understood from context. In this sentence, posao usually means a job in a general sense.

sin is in the nominative case because it is the subject of the sentence, the person doing the action.

In Moj sin traži posao:

• Moj sin = the subject
• traži = the verb
• posao = the object

So sin stays in nominative singular.

traži is the 3rd person singular present tense of the verb tražiti.

So:

• ja tražim = I am looking for
• ti tražiš = you are looking for
• on / ona / ono traži = he / she / it is looking for

Since sin means he, the verb must be traži.

tražiti most often means:

• to look for
• to search for
• sometimes to ask for or to seek

In this sentence, with posao, it clearly means to look for / seek:

• traži posao = is looking for a job

So the meaning is about job-hunting, not physically searching for a lost object.

Yes, the direct object does change case: it is in the accusative.

But here, posao is a masculine inanimate noun, and for many masculine inanimate nouns, the accusative singular is the same as the nominative singular.

So:

• nominative: posao
• accusative: posao

That is why the object still appears as posao.

A useful contrast:

• Vidim sto. = I see the table.
  (sto is masculine inanimate, so accusative = nominative)

But with masculine animate nouns, accusative is usually different:

• Vidim sina. = I see the son / my son.
  (sin is animate, so accusative changes)

Because posao is one of those nouns whose stem changes in some forms.

You will often see:

• nominative singular: posao
• genitive singular: posla
• dative singular: poslu

So the -ao in the dictionary form does not stay the same in every case.

That is completely normal and something you just get used to with certain Serbian nouns.

The letter ž is pronounced like the s in measure or the g in genre for many English speakers.

So:

• traži sounds roughly like TRAH-zhee

A rough guide to the whole sentence:

• Moj ≈ moy
• sin ≈ seen
• traži ≈ TRAH-zhee
• posao ≈ POH-sah-oh

These are only approximations, but they are helpful at the beginning.

Yes, it is a very natural and neutral word order.

It follows the common pattern:

• Subject + Verb + Object

So:

• Moj sin = subject
• traži = verb
• posao = object

Serbian word order is fairly flexible, though, and changing it can shift emphasis. For example:

• Posao traži moj sin. = It is my son who is looking for a job / A job is what my son is looking for.

But the original sentence is the plain, standard way to say it.

Yes, you could.

• Sin traži posao. = The son / a son is looking for a job, depending on context
• Moj sin traži posao. = My son is looking for a job

Adding moj makes the relationship explicit. Without it, the sentence is more general and depends more on context.

Usually it means a job in a general sense.

So tražiti posao often means:

• to look for work
• to look for a job
• to job-hunt

It does not usually imply one specific job unless the context makes that clear.

Yes. Serbian often allows the subject pronoun to be omitted because the verb form already shows the person and number.

For example:

• Traži posao. = He / she is looking for a job, depending on context

But in your sentence, Moj sin is kept because the speaker wants to say specifically my son.

So Serbian often drops pronouns like on or ona, but it does not drop important noun subjects when they carry the main information.