Hranim psa i četkam krzno.

Because pas is the nominative singular form, the basic dictionary form meaning dog.

In this sentence, the dog is the direct object of hranim (I feed), so Serbian uses the accusative case. For the masculine animate noun pas, the accusative singular is psa.

• pas = dog as subject
• psa = dog as direct object

So:

• Pas jede. = The dog is eating.
• Hranim psa. = I am feeding the dog.

Because krzno is a neuter inanimate noun, and in Serbian the nominative and accusative singular are often the same for neuter nouns.

So even though krzno is also a direct object, its form does not change:

• krzno = nominative singular
• krzno = accusative singular

That is why you get:

• četkam krzno = I brush the fur

This is different from pas, which changes because it is masculine animate.

Serbian usually drops the subject pronoun when it is clear from the verb ending.

Both verbs already show first person singular:

• hranim = I feed / I am feeding
• četkam = I brush / I am brushing

So ja is not necessary.

You could say:

• Ja hranim psa i četkam krzno.

But that would usually add emphasis, as in:

• I’m the one feeding the dog and brushing the fur.

They can mean both, depending on context.

Serbian present tense often covers both the English simple present and present continuous:

• Hranim psa can mean I feed the dog or I am feeding the dog
• Četkam krzno can mean I brush the fur or I am brushing the fur

Context tells you whether the action is:

• habitual: something you do regularly
• happening right now

Because Serbian has no articles like English a/an/the.

So:

• psa can mean a dog or the dog
• krzno can mean fur, the fur, or sometimes a fur coat in other contexts, though here it clearly means fur

Serbian relies on context to show whether something is definite or indefinite.

It is grammatically correct, but many learners notice that it feels a little less specific on its own.

If you mean the dog’s fur, Serbian often makes that clearer with a dative pronoun or noun:

• Hranim psa i četkam mu krzno. = I feed the dog and brush its fur.
• Hranim psa i četkam psu krzno. = I feed the dog and brush the dog’s fur.

The original sentence is understandable, but in isolation a native speaker might often prefer mu krzno if the fur belongs to the dog.

The infinitives are:

• hraniti = to feed
• četkati = to brush

The forms in the sentence are first person singular present tense:

• hranim = I feed / I am feeding
• četkam = I brush / I am brushing

A few related forms:

For hraniti:

• hranim = I feed
• hraniš = you feed
• hrani = he/she/it feeds

For četkati:

• četkam = I brush
• četkaš = you brush
• četka = he/she/it brushes

Yes, both hraniti and četkati are imperfective verbs, and that matters.

Imperfective verbs are used for:

• ongoing actions
• repeated actions
• habitual actions
• general present-tense statements

That fits this sentence well, because feeding and brushing are presented as actions in progress or routine actions.

In Serbian, a perfective verb usually does not describe an ongoing present action the way English does. Its present form often has a future meaning instead.

So here, imperfective is the natural choice.

No, Serbian word order is fairly flexible, because case endings help show what each noun is doing.

The given version:

• Hranim psa i četkam krzno.

is a neutral, straightforward order.

But other orders are possible for emphasis:

• Psa hranim i četkam krzno. = emphasizes the dog
• Krzno četkam. = emphasizes the fur

Even though word order can move, the original sentence sounds like the most neutral way to say it.

Yes, a few sounds may feel unusual to an English speaker:

• č is pronounced like ch in chop
• psa begins with a real ps cluster, so pronounce both consonants
• r in krzno is a tapped or rolled Serbian r

A rough guide:

• hranim ≈ HRA-neem
• psa ≈ psah
• četkam ≈ CHET-kam
• krzno ≈ KRZ-no

Also, Serbian spelling is very phonetic, so words are usually pronounced much the way they are written.