Najpierw zapisuję stałe wydatki.

Najpierw means first, first of all, or to begin with in a sequence of actions.

In this sentence, it signals that this is the first step:

• Najpierw zapisuję stałe wydatki. = First, I write down the fixed expenses.

It is a very common word when describing a process:

• Najpierw robię kawę, potem siadam do pracy. = First I make coffee, then I sit down to work.

Polish often omits subject pronouns when they are not needed.

The verb zapisuję already tells you the subject is I, because the ending -ę / -uję marks 1st person singular.

So:

• (Ja) zapisuję = I write down

Adding ja is possible, but it usually adds emphasis or contrast:

• Ja zapisuję stałe wydatki, a on liczy dochody.
  = I write down the fixed expenses, and he calculates the income.

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

Zapisuję is the 1st person singular present tense form of zapisywać.

So it means:

• I write down
• I am writing down
• sometimes I enter/record (depending on context)

The exact English translation depends on context, but the Polish form itself is just present tense.

A short conjugation of zapisywać:

• ja zapisuję = I write down
• ty zapisujesz = you write down
• on/ona/ono zapisuje = he/she/it writes down
• my zapisujemy = we write down
• wy zapisujecie = you (plural) write down
• oni/one zapisują = they write down

Because zapisuję is more specific.

• pisać / piszę = to write
• zapisywać / zapisuję = to write down, to note down, to record

With expenses, Polish usually prefers something like zapisuję wydatki, because the idea is not just writing in general, but recording them in a list, notebook, spreadsheet, budget, and so on.

Compare:

• Piszę list. = I’m writing a letter.
• Zapisuję wydatki. = I’m writing down / recording expenses.

It is imperfective.

The infinitive here is zapisywać (imperfective), which is used for:

• ongoing actions
• repeated/habitual actions
• describing a procedure or process

That fits this sentence well, because it can mean something like:

• First, I write down the fixed expenses
  as part of a routine or explanation of how you do budgeting.

The related perfective verb is zapisać. Its 1st person singular future-like form is:

• zapiszę = I will write down / I’ll note down

So:

• Najpierw zapisuję stałe wydatki. = a routine/process description
• Najpierw zapiszę stałe wydatki. = First, I’ll write down the fixed expenses (one completed action in the future)

Stałe wydatki means fixed expenses, regular expenses, or recurring expenses.

• stałe = fixed, constant, regular
• wydatki = expenses

This usually refers to expenses that come back regularly, for example:

• rent
• electricity bills
• phone bill
• subscriptions
• loan payments

So it is not just any expenses, but the ones that are predictable and recurring.

Because stałe has to agree with wydatki.

In Polish, adjectives agree with the noun in:

• gender
• number
• case

Here, wydatki is plural, so the adjective must also be plural.
For this kind of plural noun, the correct adjective form is stałe.

So:

• stały wydatek = a fixed expense (singular)
• stałe wydatki = fixed expenses (plural)

That is why stały wydatki would be incorrect.

It is the accusative case, because it is the direct object of the verb zapisuję.

You are writing down what?

• stałe wydatki

However, this is one of those places where the form looks the same as the nominative.

For non-masculine-personal plural nouns, the accusative plural is often the same as the nominative plural:

• nominative: stałe wydatki
• accusative: stałe wydatki

So even though the case is accusative, the words do not visibly change here.

The singular is wydatek.

So:

• wydatek = expense
• wydatki = expenses

Examples:

• To duży wydatek. = That is a big expense.
• Mam duże wydatki. = I have big expenses.

In budgeting contexts, the plural wydatki is very common, because people usually talk about a whole set of expenses rather than just one.

Yes. Polish word order is more flexible than English word order.

The original sentence:

• Najpierw zapisuję stałe wydatki.

is very natural and neutral.

Other possible orders include:

• Zapisuję najpierw stałe wydatki.
• Stałe wydatki zapisuję najpierw.

These are grammatically possible, but they may sound slightly different in emphasis:

• putting najpierw first emphasizes the sequence: first
• putting stałe wydatki first emphasizes what you write down

For a learner, the original version is a very good, natural choice.

A rough English-friendly guide is:

Najpierw zapisuję stałe wydatki
≈ NAI-pyerv zah-pee-SOO-yeh STAH-weh vih-DAT-kee

A few useful pronunciation notes:

• j in Polish sounds like English y in yes
• w sounds like English v
• ł sounds like English w
• ę at the end of zapisuję is nasal, but in normal speech it is often heard roughly like yeh with a slight nasal quality

So:

• najpierw → roughly NAI-pyerv
• zapisuję → roughly zah-pee-SOO-yeh
• stałe → roughly STAH-weh
• wydatki → roughly vih-DAT-kee

If you want to sound natural, the main stress falls on the second-to-last syllable in each word:

• najPIERW
• zapiSUję
• STAłe
• wyDATki