Danas sam umorna, jer nisam dobro spavala i želim da se odmaram kod kuće.

Serbian often drops personal pronouns when they are not needed, because the verb form already shows the person.

Here, sam tells you it is 1st person singular: I am.

So:

• Danas sam umorna = Today I am tired
• Ja sam danas umorna also works, but ja is usually added only for emphasis, contrast, or clarity.

A native speaker will very often leave out ja unless they want to stress I specifically.

Because sam is a clitic in Serbian. Clitics usually go in the second position in the clause.

So in:

• Danas sam umorna

the first element is Danas, and the clitic sam comes right after it.

This is very normal Serbian word order.

Compare:

• Danas sam umorna = Today I am tired
• Ja sam danas umorna = I am tired today

In both cases, sam appears early, in its usual clitic position.

Those endings show that the speaker is feminine singular.

• umorna = tired, said by a woman
• spavala = slept, said by a woman

If the speaker were male, it would be:

• Danas sam umoran, jer nisam dobro spavao...

So Serbian often tells you the speaker’s gender through adjectives and past-tense forms.

The negative forms of biti in the present tense are special fixed forms:

• nisam = I am not
• nisi = you are not
• nije = he/she/it is not
• nismo = we are not
• niste = you plural/formal are not
• nisu = they are not

So you do not normally say ne sam. The correct form is nisam.

In this sentence:

• nisam dobro spavala = I did not sleep well

Because dobro here is an adverb, not an adjective.

It modifies the verb spavala:

• dobro spavati = to sleep well

Compare:

• dobar san = a good sleep/dream → adjective
• dobro spavati = to sleep well → adverb

English also does this difference:

• good = adjective
• well = adverb

So dobro is correct because it means well.

Because each part describes a different time relationship:

• sam umorna = present state → I am tired now
• nisam dobro spavala = past cause → I didn’t sleep well
• želim = present wish → I want

So the sentence works like this:

• I am tired now
• because earlier I didn’t sleep well
• and now I want to rest at home

This mix of present and past is completely natural.

Because Serbian very often uses da + present tense where English uses an infinitive.

So:

• želim da se odmaram literally looks like I want that I rest
• but it simply means I want to rest

This is one of the most important Serbian patterns to get used to:

• želim da idem = I want to go
• moram da radim = I have to work
• volim da čitam = I like to read

An infinitive exists in Serbian, but in everyday Serbian, especially in this kind of sentence, da + present is very common and natural.

Here se is part of the verb odmarati se.

So the full verb is:

• odmarati se = to rest, to relax

In many cases, se does not translate neatly as myself in English. It is just part of how the verb is used.

So you should learn it as a whole unit:

• odmarati se = to rest
• Ja se odmaram = I am resting

If you remove se, the sentence becomes ungrammatical or changes the verb’s meaning.

This is about aspect, which is very important in Serbian.

• odmarati se = imperfective → resting, relaxing, being in the process/habit
• odmoriti se = perfective → to get rested, to rest up, to complete the action

In your sentence:

• želim da se odmaram emphasizes the activity/state of resting

If you said:

• želim da se odmorim

that would sound more like I want to rest up / recover / get some rest.

Both can be possible in the right context, but they are not exactly the same nuance.

Kod kuće is a very common Serbian expression meaning at home.

The word kod normally takes the genitive case, so kuća becomes kuće:

• kuća = house, home
• kod kuće = at home

So this is both:

1. an idiomatic expression you should memorize as a chunk, and
2. a case pattern, because kod requires genitive.

Not if you simply mean at home.

• kod kuće = at home
• u kući = in the house, inside the building

So the difference is:

• Ostajem kod kuće = I’m staying home
• Ona je u kući = She is in the house

English sometimes uses home and house differently, and Serbian does too. In this sentence, kod kuće is the natural choice.

Because jer introduces a subordinate clause of reason, similar to because.

In Serbian punctuation, it is normal to put a comma before jer:

• Danas sam umorna, jer nisam dobro spavala...

So the comma separates the main clause from the explanatory clause.

No. Jer is common and natural, but it is not the only option.

Other possibilities include:

• zato što = because
• sometimes pošto = since, because

Examples:

• Umorna sam, jer nisam spavala.
• Umorna sam, zato što nisam spavala.

These are very similar in meaning here. In your sentence, jer is perfectly normal and natural.