Learn Hardcore Indonesian: Pengeluaran rumah tangga kami berkurang setelah anggaran diubah. - Our household expenses decreased after the budget was changed.

The subject is pengeluaran rumah tangga kami (our household expenses). The verb is berkurang (decreased). The time clause is setelah anggaran diubah (after the budget was changed).

Why is kami used here instead of kita? Could I use kita?

kami = we/us (excluding the listener).
kita = we/us (including the listener).
If you want to include the person you’re talking to as part of the household, you can say pengeluaran rumah tangga kita. Otherwise, kami is correct.

Does the sentence tell us who changed the budget? How can I say it explicitly?

No, anggaran diubah is passive without an agent. To make the agent explicit:

anggaran kami diubah oleh kami/manajer keuangan (our budget was changed by us/the finance manager).
Or use active voice: kami mengubah anggaran (we changed the budget).

What’s the difference between diubah, berubah, and mengubah?

diubah = was changed (passive; someone changed it).
berubah = changed/altered on its own (intransitive).
mengubah = to change (active, transitive; someone changes something).
Examples: Anggaran diubah kemarin. / Anggaran berubah (it changed). / Kami mengubah anggaran.

Why is it berkurang and not mengurangi?

berkurang is intransitive: something decreases by itself.
mengurangi is transitive: someone reduces something.

Intransitive: Pengeluaran ... berkurang.
Transitive: Kami mengurangi pengeluaran.

Do I need a past marker like sudah or telah?

Not required. Setelah already orders the events in time. You can add them for emphasis or formality:

Pengeluaran ... telah berkurang setelah anggaran diubah.
Pengeluaran ... berkurang setelah anggaran sudah diubah.

Can I move the time clause to the front?

Yes. Add a comma after it:

Setelah anggaran diubah, pengeluaran rumah tangga kami berkurang.

Should it be anggaran kami diubah to show it’s our budget?

If you want to be explicit, yes: setelah anggaran kami diubah. In context, anggaran diubah is often understood to mean the relevant budget (likely yours), but adding kami removes ambiguity.

What does rumah tangga mean exactly? Is it the same as keluarga?

rumah tangga = household (the domestic unit/operations).
keluarga = family (the people/relatives).
So pengeluaran rumah tangga is household spending; pengeluaran keluarga can work but focuses on the family as people rather than the household’s operations.

Are there other natural ways to say “household expenses”?

Yes:

pengeluaran rumah tangga (most standard)
biaya rumah tangga
belanja rumah tangga
biaya hidup (cost of living; broader)

What are some synonyms for berkurang here, and do they differ in tone?

menurun (decline; neutral/formal)
turun (go down; common/informal)
menyusut (shrink; slightly technical)
merosot (plummet/slump; stronger negative) Example: Pengeluaran ... menurun/turun setelah anggaran diubah.

Is pengeluaran countable? How do I say “individual expenses”?

As a mass noun, pengeluaran means overall expenditure. For individual items, say pengeluaran-pengeluaran (reduplication) or use biaya/pos pengeluaran:

Kami memangkas pos-pos pengeluaran.

Is rumah tangga written as one word or two?

Two words: rumah tangga. Writing rumahtangga is nonstandard.

Is setelah the only option? What about casual speech?

Synonyms:

sesudah (formal/neutral)
sehabis (neutral/slightly informal)
Very casual: habis/abis (spoken).
Example: Pengeluaran ... berkurang sesudah/sehabis anggaran diubah.

How would I say this in active voice to avoid the passive?

Setelah kami mengubah anggaran, pengeluaran rumah tangga kami berkurang.
This explicitly shows who changed the budget.

What are the word-formation pieces here?

pengeluaran = peN- + keluar
- -an → “the outgo(ing)” → expenditure.
berkurang = ber- + kurang → to decrease/be reduced.
diubah = di- + ubah → be changed.
anggaran = budget (also bujet informally).