Želim da udomim mačku.

Here, da introduces a subordinate verb clause. In sentences like this, it often does the job that English to does in to adopt.

So Serbian says:

Želim da udomim...

where English says:

I want to adopt...

It is not best understood as a literal word-for-word that here.

Because that is a very common Serbian pattern after verbs of wanting, planning, trying, and similar ideas.

So Želim da udomim mačku is the normal, natural Serbian structure.

The infinitive udomiti exists, but in modern Serbian, da + present tense is usually more natural in this kind of sentence.

Because the subject of that second verb is still I.

In Serbian, after da, the verb is normally fully conjugated:

• Želim da udomim mačku = I want to adopt a cat
• Želim da on udomi mačku = I want him to adopt a cat

So udomim matches the speaker.

Udomim is the 1st person singular present tense form of udomiti.

Even though it is a present-tense form, in a sentence like this it refers to a desired future action, not something happening right now.

Udomiti is perfective.

That means it presents the action as a complete whole, with a result: the cat ends up having a home.

That fits well here, because the speaker wants to carry out one completed action of adoption.

The imperfective partner is udomljavati, which is used more for repeated, ongoing, or habitual action.

Because mačku is the accusative singular form, and here the cat is the direct object of the verb.

The dictionary form is mačka, but many feminine nouns ending in -a change to -u in the accusative singular:

• mačka → mačku

No. Serbian does not have articles like a or the.

So mačku by itself can mean a cat or the cat, depending on context.

English speakers often expect a separate word here, but Serbian simply does not use one.

Literally, mačka is female cat, while mačak is male cat.

But in everyday Serbian, mačka is also very commonly used in a general sense for cat, especially when the sex is not important.

So in this sentence, it sounds completely natural.

Not exactly.

For animals, udomiti is especially natural because it focuses on giving an animal a home.

Usvojiti can mean adopt too, but it is more strongly associated with formal or legal adoption, especially of children.

So for a cat, udomiti is a very good choice.

Yes. Serbian word order is fairly flexible because case endings show what each noun is doing.

The given sentence is a neutral, natural order:

Želim da udomim mačku.

You can change the order for emphasis, for example:

Mačku želim da udomim.

That puts extra focus on the cat. But for a learner, the original order is the safest default.

A rough pronunciation is:

ZHE-lim da oo-DO-mim MAHCH-koo

A few useful sounds:

• ž sounds like the s in measure
• č sounds like ch in church
• u is like oo in food

Usually, no.

Želim da udomim mačku is already natural and complete.

You would add jednu only if you want to stress one:

Želim da udomim jednu mačku, ne dve.
= I want to adopt one cat, not two.