Kada nemam plan za vikend, volim da ostanem kod kuće i čitam.

Yes. Kada means when.

In everyday Serbian, kad is very common and often sounds more natural in casual speech:

• Kada nemam plan za vikend...
• Kad nemam plan za vikend...

Both are correct.
Very broadly:

• kada = a bit more neutral or formal
• kad = more conversational

Because Kada nemam plan za vikend is a subordinate clause, and it is followed by the main clause:

• Kada nemam plan za vikend,
• volim da ostanem kod kuće i čitam.

Serbian usually separates this kind of dependent clause with a comma. This is more regular in Serbian than in English.

Because Serbian often drops subject pronouns when the verb ending already shows who the subject is.

Here the verb forms tell us it is I:

• nemam = I do not have
• volim = I like
• ostanem = I stay / remain
• čitam = I read

So ja is unnecessary unless you want emphasis:

• Ja volim da ostanem kod kuće... = I like to stay home...

That would sound more contrastive or emphatic.

Nemam is the 1st person singular present form of imati = to have.

So:

• imam = I have
• nemam = I do not have

Serbian often forms negation by attaching ne- to the verb:

• imam → nemam
• volim → ne volim

Because Serbian often uses the singular in places where English might naturally use either singular or plural.

Here nemam plan za vikend means something like:

• I don’t have a plan for the weekend
• I don’t have any plans for the weekend

If you want to stress multiple plans, you could say:

• nemam planove za vikend

But nemam plan za vikend is perfectly natural and idiomatic.

Here za vikend means for the weekend or, depending on context, on/during the weekend.

After za, vikend is in the accusative case. But vikend is an inanimate masculine noun, so its accusative singular looks the same as the nominative singular:

• nominative: vikend
• accusative: vikend

So even though the form does not change, the case is accusative.

Also note that Serbian has more than one way to express on weekends / during the weekend, for example:

• za vikend = for/on the weekend
• vikendom = on weekends, on weekends generally

In Serbian, it is very common to use da + present tense where English uses to + infinitive.

So:

• volim da ostanem = I like to stay
• literally: I like that I stay/remain

This is completely normal Serbian grammar.

An infinitive is much less central in modern Serbian than in English. In many situations, especially after verbs like voleti, morati, želeti, moći, Serbian prefers da + present.

So for a learner, a good rule is:

• English to stay
• Serbian often da ostanem

This is an aspect question.

• ostati → perfective
• ostajati → imperfective

Here ostanem comes from ostati, and it sounds natural because the idea is to end up staying / to remain at home as a complete choice or result.

By contrast, ostajem would usually suggest an ongoing or repeated process and would sound less natural in this sentence.

So:

• volim da ostanem kod kuće = I like to stay home / remain at home
• ostanem presents the action as a whole

This is one of those places where Serbian uses aspect in a way that does not map perfectly onto English.

Kod kuće is the normal expression for at home.

Literally, kod often means something like at/by someone’s place, and it takes the genitive case.
That is why you get:

• kod kuće

Here kuće is the genitive singular of kuća.

Important contrast:

• kod kuće = at home
• u kući = in the house

So if you want the general idea of being at home, use kod kuće.
If you specifically mean physically inside the building, u kući is possible.

It is genitive singular.

The preposition kod requires the genitive case, so:

• kuća = house/home
• kuće = of the house / at home, depending on the expression

This is why the phrase is:

• kod kuće

not kod kuća.

Because čitati is an imperfective verb, and here it refers to a general activity: reading.

Compare:

• čitati = to read, to be reading, to read in general
• pročitati = to read through, to finish reading

In this sentence, the speaker is not talking about finishing a specific book. They are talking about a typical leisure activity. That is why čitam is the natural choice.

So:

• volim da ... čitam = I like to read

If you said pročitam, it would suggest finishing something, which is not the idea here.

Because Serbian can omit the repeated da when two verbs are coordinated and belong to the same structure.

So these are both possible:

• volim da ostanem kod kuće i čitam
• volim da ostanem kod kuće i da čitam

The second version repeats da, which can make the structure a bit more explicit.
The version without the second da is very natural and common.

It uses the present tense, but the meaning is habitual/general, not necessarily happening right now.

That is very normal in Serbian.

• Kada nemam plan za vikend... = When I don’t have a plan for the weekend...
• volim da ostanem... = I like to stay...
• čitam = I read / to read

So the present tense here expresses a repeated habit or general preference.

Yes. That version is also correct.

It differs slightly in style:

• kad instead of kada = more conversational
• planove instead of plan = explicitly plural, closer to English plans
• repeating da before čitam = a bit more explicit

So your version is grammatical and natural.
The original sentence is just a slightly more compact way to say the same thing.