Příští týden se stěhujeme do nového bytu, proto už dnes balím knihy a kabáty.

Stěhovat se is the normal Czech verb for to move house / to relocate.

• stěhovat = to move something
• stěhovat se = to move oneself, i.e. to change where you live

So:

• Stěhujeme se. = We are moving.

In this sentence, se is part of the verb, not a separate word with its own independent meaning like English self. Many Czech verbs are reflexive in this way, and you simply learn them as a unit:

• učit = to teach

• učit se = to learn

• stěhovat = to move something
• stěhovat se = to move house

So here se stěhujeme means we are moving.

Czech often uses the present tense to talk about a planned or expected future action, especially when the time is clearly stated.

So:

• Příští týden se stěhujeme. = We are moving next week.

This is very similar to English:

• We are moving next week.

Because příští týden already makes the future time clear, Czech does not need a special future form here.

This is a very natural shift in subject.

• se stěhujeme = we are moving
• balím = I am packing

The meaning is:

• We are moving next week, so I’m already packing books and coats today.

So the first part is about the whole household or group, while the second part focuses only on the speaker.

If the speaker wanted to say that everyone is packing, it would be:

• ...proto už dnes balíme knihy a kabáty. = ...so we are already packing books and coats today.

Because do takes the genitive case when it means movement into something.

Here the basic noun is:

• byt = apartment / flat

After do, it becomes genitive singular:

• do bytu = into the apartment

The adjective must match it:

• nový byt = a new apartment
• do nového bytu = into a new apartment

So both words change because of the preposition do.

It is genitive singular.

Breakdown:

• byt → bytu
• nový → nového

Why genitive? Because do requires it when expressing motion toward the inside of something.

So:

• do nového bytu = into a new apartment

They are the direct objects of balím (I am packing), so they are in the accusative plural.

• kniha → knihy
• kabát → kabáty

In this sentence:

• balím knihy a kabáty = I am packing books and coats

A useful point:

For many inanimate masculine and feminine nouns, the accusative plural often looks the same as the nominative plural.

So:

• knihy can mean books as subject or object depending on context
• kabáty can also be nominative plural or accusative plural

Here they are objects because they follow balím.

Because Czech often uses time expressions without a preposition to mean when something happens.

• příští týden = next week
• příští měsíc = next month
• příští rok = next year

This works much like English, where you also say:

• next week not
• in next week

So Příští týden se stěhujeme is completely normal.

Proto means therefore, that’s why, or so.

It connects the two parts logically:

• We are moving next week, therefore / so I’m already packing today.

In Czech, proto is a common way to show result or consequence.

Compare:

• Stěhujeme se, proto balím. = We’re moving, so I’m packing.

It is slightly more explicit and formal than just loosely joining the clauses.

Because Czech separates these two clauses with a comma, and proto introduces the result of the first clause.

So the structure is:

• Příští týden se stěhujeme, proto už dnes balím...

This comma is standard Czech punctuation.

Už dnes means already today or more naturally in English, as early as today / already today.

It emphasizes that the speaker is not waiting until next week:

• We’re moving next week, so I’m already packing today.

Word-by-word:

• už = already
• dnes = today

The placement before balím is natural because it highlights the time adverbial before the action:

• proto už dnes balím...

Because Czech verb endings usually already show the subject.

• stěhujeme = we move / we are moving
• balím = I pack / I am packing

So:

• (my) se stěhujeme
• (já) balím

The pronouns my and já are usually omitted unless you want extra emphasis or contrast.

For example:

• Já balím knihy, ale on nic nedělá. = I’m packing books, but he isn’t doing anything.

In your sentence, no pronouns are needed.

Balím is a present-tense form of an imperfective verb, and here it describes an action that is happening already now.

• balím = I am packing / I pack

Because the sentence says už dnes, the packing is happening today, so present tense is exactly right.

If the meaning were future, you could say:

• Budu balit zítra. = I’ll be packing tomorrow.

But in your sentence the speaker has already started, so balím is the natural choice.

The sentence’s word order is natural, but Czech word order is fairly flexible.

Original:

• Příští týden se stěhujeme do nového bytu, proto už dnes balím knihy a kabáty.

This order sounds neutral and clear:

1. time first
2. action of moving
3. result
4. time emphasis
5. packing action

You could rearrange parts for emphasis, for example:

• Už dnes balím knihy a kabáty, protože se příští týden stěhujeme do nového bytu.

That means basically the same thing, but the focus shifts slightly toward the packing.

So the original word order is not the only possible one, but it is very natural.

Not always. Byt usually means apartment or flat, depending on the variety of English.

• American English: apartment
• British English: flat

So:

• do nového bytu = into a new apartment / flat

The Czech word itself does not distinguish between those two English choices.