Dziś wieczorem chcę odpocząć w domu.

Because Polish often drops subject pronouns when they are clear from the verb form.

• chcę = I want
• So ja is not necessary

You could say Ja dziś wieczorem chcę odpocząć w domu, but that sounds more emphatic, as if you are stressing I.

Both mean today.

• dziś is a little shorter and slightly more compact in style
• dzisiaj is also completely normal and very common

So these are both fine:

• Dziś wieczorem chcę odpocząć w domu.
• Dzisiaj wieczorem chcę odpocząć w domu.

For a learner, it is safest to treat them as near-equivalents in everyday speech.

Because Polish uses wieczorem to mean in the evening / tonight, as an adverbial time expression.

• wieczór = evening as a noun
• wieczorem = in the evening

This is a very common pattern in Polish. The form wieczorem is historically connected with the instrumental case, but for learners the main thing is to remember it as the standard expression for time.

Compare:

• To był miły wieczór. = It was a nice evening.
• Spotkamy się wieczorem. = We’ll meet in the evening.

Yes, literally it is close to today evening, but in natural English it is usually translated as:

• this evening
• tonight

Polish often combines dziś + a time word:

• dziś rano = this morning
• dziś po południu = this afternoon
• dziś wieczorem = this evening / tonight

This is normal in Polish. After chcę (I want), Polish uses the infinitive of the second verb.

• chcę odpocząć = I want to rest
• chcę iść = I want to go
• chcę spać = I want to sleep

So the pattern is:

chcę + infinitive

Very similar to English want + to + verb, except Polish does not need a separate word like to here.

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

• odpocząć = perfective
• odpoczywać = imperfective

In this sentence, odpocząć suggests having a complete rest / taking a rest / resting for a bit as a whole action. With chcę, Polish often uses the perfective when the speaker means a single intended action.

So:

• chcę odpocząć = I want to rest, have some rest
• chcę odpoczywać = I want to be resting / I want to rest regularly or over a period of time

For this sentence, odpocząć is the most natural choice.

It is the locative case.

The preposition w often takes the locative when it means in or at a place.

• dom = house/home
• w domu = in the house / at home

So here:

• base form: dom
• locative form: domu

This is why you do not say w dom.

They mean different things:

• w domu = at home / in the house
• do domu = to home / homeward / to the house

So:

• Chcę odpocząć w domu. = I want to rest at home.
• Chcę iść do domu. = I want to go home.

A useful shortcut:

• w often answers where?
• do often answers to where?

It can mean both, depending on context.

• at home = focusing on location in a general sense
• in the house = focusing more literally on being inside the building

In your sentence, at home is the most natural English translation.

Yes. Polish word order is more flexible than English, because endings carry a lot of grammatical information.

These are all possible:

• Dziś wieczorem chcę odpocząć w domu.
• Chcę dziś wieczorem odpocząć w domu.
• W domu chcę dziś wieczorem odpocząć.

But they do not all sound identical. The usual effect is:

• the first part of the sentence often gets more focus
• the last part can also sound more emphasized

Your original sentence is very natural and neutral.

Approximate help for an English speaker:

• dziś sounds roughly like jeesh but shorter and softer
• chcę is tricky; the beginning chc- is a consonant cluster, and the whole word is roughly like h-tsen with a nasal vowel at the end
• odpocząć is roughly ot-po-chonch, though that is only approximate

A few pronunciation notes:

• dz / dź / dzi sounds are important in Polish and are softer than many English sounds
• ę is a nasal vowel, though in real speech it often changes depending on what follows
• cz sounds like ch in chocolate

If you want to sound natural, it helps to learn the sentence in chunks:

• Dziś wieczorem
• chcę odpocząć
• w domu

Yes, that is grammatically fine, but it means something slightly different.

• odpocząć = rest
• zrelaksować się = relax

Both are natural, but odpocząć is simpler, more basic, and very common.
Your original sentence sounds completely normal in everyday Polish.