Veterinar je rekao da naš ljubimac mora na cijepljenje sljedeći tjedan.

Because Croatian normally forms the past tense with:

• a present-tense auxiliary from biti = sam, si, je, smo, ste, su
• plus the l-participle

So:

• je = the auxiliary
• rekao = the past participle-like form of reći = to say

Together, je rekao means said.

In this sentence, Veterinar je rekao = The vet said.

Because je is a clitic, and Croatian clitics usually go in second position in the clause.

So in:

• Veterinar je rekao ...

the first element is Veterinar, and the clitic je comes right after it.

This is standard Croatian word order. English speakers often expect the two words of the past tense to stay together more rigidly, but Croatian is more flexible.

You may also hear other word orders in different contexts, for example:

• Rekao je veterinar ...

but Veterinar je rekao ... is completely normal and neutral here.

Because rekao agrees with the speaker’s grammatical gender and number.

Here, veterinar is masculine singular, so the participle is also masculine singular:

• veterinar je rekao = a male vet said

If the vet were female, you would say:

• veterinarka je rekla

So the participle changes:

• rekao = masculine singular
• rekla = feminine singular
• rekli = masculine plural / mixed group
• rekle = feminine plural

Da introduces a subordinate clause, very often corresponding to English that.

So:

• Veterinar je rekao da ...
  means
• The vet said that ...

After verbs like reći (to say), misliti (to think), znati (to know), vidjeti (to see), Croatian usually uses da to introduce the content of what was said, thought, known, and so on.

Usually no, or at least not in normal standard Croatian.

In English, you can say both:

• The vet said that our pet has to go ...
• The vet said our pet has to go ...

In Croatian, da is normally kept:

• Veterinar je rekao da naš ljubimac mora ...

Leaving it out would sound unnatural in standard usage.

Because naš ljubimac is the subject of the subordinate clause:

• da naš ljubimac mora ... = that our pet has to ...

Subjects are normally in the nominative case.

So:

• naš = nominative masculine singular
• ljubimac = nominative masculine singular

If ljubimac were an object, then a different case might be needed, such as ljubimca.

Not necessarily.

Ljubimac is a masculine noun grammatically, but it can be used generically for pet without focusing on biological sex.

So it can mean simply pet.

If you specifically want to refer to a female pet, you can say:

• ljubimica

But in many everyday sentences, ljubimac works as a general word for a pet.

Mora is the 3rd person singular of morati = must / have to.

So:

• ja moram = I must
• ti moraš = you must
• on/ona/ono mora = he/she/it must

Here:

• naš ljubimac mora ... = our pet must / has to ...

In natural English, has to is often the best translation in this kind of sentence.

Here na is part of a very common Croatian pattern used for going to an event, procedure, appointment, or activity.

So:

• ići na pregled = go for an examination
• ići na operaciju = go for surgery
• ići na cijepljenje = go for vaccination

In this sentence:

• mora na cijepljenje
  literally something like
• must to vaccination

but natural English is:

• has to go for vaccination
• has to go get vaccinated
• has to go for a vaccination

So this is not just the basic spatial meaning of na = on. It is an idiomatic and very common use.

After na expressing movement or destination, Croatian normally uses the accusative.

So in:

• na cijepljenje

cijepljenje is accusative singular.

However, cijepljenje is a neuter noun, and for many neuter nouns the nominative and accusative forms look the same. That is why you do not see a visible ending change here.

So:

• nominative: cijepljenje
• accusative: cijepljenje

Same form, different function.

Cijepljenje is a noun meaning vaccination.

It refers to the procedure itself, not the vaccine as an object. So:

• cjepivo = vaccine
• cijepljenje = vaccination / vaccinating

That is why na cijepljenje is best understood as for vaccination or to get vaccinated.

Because Croatian often uses a bare case form for time expressions, especially with words like today, tomorrow, last week, next month, and so on.

So:

• sljedeći tjedan = next week
• prošli tjedan = last week
• ovaj mjesec = this month

You do not need a preposition like English sometimes uses in other contexts.

A phrase like u sljedećem tjednu is possible in some special contexts, but for ordinary next week, sljedeći tjedan is the natural choice.

It is functioning as an accusative of time, a common way to express when something happens.

So the sentence says the pet has to go for vaccination next week.

A useful detail: tjedan is masculine inanimate, and in the singular its nominative and accusative look the same:

• nominative: tjedan
• accusative: tjedan

That is why the form does not visibly change.

Likewise, sljedeći here matches that form.

Yes, you can.

Both are common:

• sljedeći tjedan
• sljedećeg tjedna

They are very close in meaning. For many learners, the simplest approach is:

• sljedeći tjedan = a very straightforward next week
• sljedećeg tjedna = also next week, often with a slightly more during next week / at some point next week feel

In everyday use, both may be heard, but sljedeći tjedan is a very easy and standard pattern to learn first.

Because in Croatian, when da introduces a content clause that is a direct complement of a verb like reći, misliti, or znati, you normally do not put a comma before it.

So this is standard:

• Veterinar je rekao da naš ljubimac mora na cijepljenje sljedeći tjedan.

This is similar to:

• Mislim da je u pravu.
• Znam da dolazi.

English punctuation habits can make learners want a comma here, but Croatian normally does not use one.

Croatian word order is fairly flexible, although some versions sound more neutral than others.

The given sentence is very natural and neutral:

• Veterinar je rekao da naš ljubimac mora na cijepljenje sljedeći tjedan.

But you could also hear:

• Veterinar je rekao da sljedeći tjedan naš ljubimac mora na cijepljenje.
• Sljedeći tjedan naš ljubimac mora na cijepljenje, rekao je veterinar.

Changing the order usually changes emphasis more than basic meaning.

In the original sentence, the emphasis is fairly balanced and conversational.

Because Croatian has no articles like English the or a/an.

So:

• veterinar can mean the vet or a vet
• ljubimac can mean the pet or a pet

Which meaning is intended comes from context.

In your sentence, context makes it clear that veterinar means the vet and naš ljubimac means our pet.

Yes, in many everyday contexts it could.

Both can mean next / upcoming:

• sljedeći tjedan
• idući tjedan

For most learners, sljedeći tjedan is a very safe and standard choice. You will also hear idući tjedan often in speech.

Just be aware that usage and stylistic preference can vary a bit by speaker and region.