Po podpisaniu umowy wrócimy do domu i zaczniemy pakować ostatnie rzeczy.

Podpisaniu is a verbal noun form built from the verb podpisać (to sign). The basic noun-like form is podpisanie (signing). After the preposition po when it means after, Polish uses the locative case, so podpisanie changes to podpisaniu.

So:

• podpisać = to sign
• podpisanie = signing
• po podpisaniu = after signing

This is a very common Polish pattern.

Because Polish does not use po + infinitive the way English can use after + -ing.

After po meaning after, Polish normally uses:

• po + noun
• or po + verbal noun

So po podpisaniu umowy literally means after the signing of the contract, and naturally corresponds to English after signing the contract.

A full clause is also possible, for example:

• kiedy podpiszemy umowę = when we sign the contract
• po tym, jak podpiszemy umowę = after we sign the contract

But po podpisaniu umowy is shorter and very natural.

This is because podpisanie is a verbal noun, and nouns that depend on verbal nouns are very often put in the genitive.

Compare:

• podpisać umowę = to sign a contract
  
  • umowę is accusative, because it is the direct object of the verb
• podpisanie umowy = the signing of a contract
  
  • umowy is genitive, because it depends on the noun podpisanie

This is a very important pattern in Polish: a verb may take an accusative object, but the corresponding verbal noun often takes a genitive complement.

Because podpisać is perfective, and here the sentence is talking about a completed action that happens before the next ones.

• po podpisaniu umowy = after the contract has been signed / after signing the contract
• po podpisywaniu umowy would suggest during/after the process of signing, which is not the intended meaning here

In this sentence, the contract needs to be fully signed first, and only then do the later actions happen. That is why the perfective idea fits best.

Because both verbs are perfective:

• wrócić = to return, to come back
• zacząć = to begin

In Polish, perfective verbs do not normally have a true present tense. Their present-looking forms are used with future meaning.

So:

• wrócimy = we will return
• zaczniemy = we will begin

Compare with the imperfective verbs:

• wracamy = we are returning / we return
• zaczynamy = we are beginning / we begin

This is one of the biggest differences between Polish and English.

Because Polish usually drops subject pronouns when the verb ending already makes the subject clear.

• wrócimy already means we will return
• zaczniemy already means we will begin

So my is unnecessary unless you want emphasis or contrast, for example:

• My wrócimy do domu, a oni zostaną.
  We will go home, and they will stay.

In a neutral sentence, leaving out my is completely normal.

After zacząć (to begin/start), Polish usually uses an imperfective infinitive to describe the activity that is starting.

So:

• zaczniemy pakować = we will start packing

This sounds natural because packing is seen as a process.

Using a perfective infinitive after zacząć is usually much less natural, because perfective verbs focus on the action as completed, while start focuses on the beginning of the action.

So in most ordinary situations:

• zacząć pakować = natural
• zacząć spakować = not normal

Because Polish uses do + genitive to show movement toward a place in many common expressions.

Here:

• dom = house/home
• genitive singular = domu
• do domu = to home / home / to the house

So:

• wrócimy do domu = we will return home

This is a very common phrase, and learners should memorize it as a unit.

In many contexts, especially with verbs like wrócić, iść, jechać, do domu is best translated simply as home.

So:

• wrócimy do domu = we will return home

It can literally mean to the house, but in everyday use it often refers to going back to one’s home, not just to a building in a neutral architectural sense.

It is the accusative plural, because it is the direct object of pakować:

• pakować co? = to pack what?
• ostatnie rzeczy = the last/remaining things

Why does it look like the nominative? Because for non-masculine-personal plurals, the nominative and accusative forms are often the same.

Here:

• rzecz = thing
• rzeczy = things
• ostatnie rzeczy = last/remaining things

The adjective ostatnie agrees with rzeczy in number, gender/class, and case.

In this context, ostatnie rzeczy usually means the last remaining things or the last few items left to pack.

It does not necessarily mean the final things in a dramatic sense. It often just means that most packing is already done, and only a few items are still left.

So a very natural English rendering is:

• the last things
• the remaining things
• the last few items

Because po podpisaniu umowy is a prepositional phrase, not a subordinate clause.

Polish normally does not put a comma after this kind of short introductory phrase:

• Po obiedzie poszliśmy na spacer.
• Po pracy wrócę do domu.
• Po podpisaniu umowy wrócimy do domu.

A comma would be expected with a full clause, for example:

• Kiedy podpiszemy umowę, wrócimy do domu.

So the lack of a comma here is standard.

Yes, Polish word order is fairly flexible, but changing it changes the focus or emphasis.

The given version is neutral:

• Po podpisaniu umowy wrócimy do domu i zaczniemy pakować ostatnie rzeczy.

You could also say:

• Wrócimy do domu po podpisaniu umowy i zaczniemy pakować ostatnie rzeczy.

This still works, but the rhythm and emphasis are a little different. Starting with Po podpisaniu umowy neatly sets the time frame first, which is very natural in Polish.

So the order can change, but the original sentence is a very normal, neutral way to say it.