Learn Hardcore Spanish (Latin America): Nos vamos a mudar a un apartamento pequeño el sábado. - We are going to move to a small apartment on Saturday.

nos is the reflexive pronoun required by the verb mudarse (to move house). It shows the action affects the subject (ourselves).
vamos is the 1st-person plural of ir, used in the periphrastic future ir a + infinitive (“going to”).
The subject pronoun nosotros is optional and omitted here. You cannot drop nos with mudarse.

Can I say Vamos a mudarnos instead of Nos vamos a mudar?

Yes. Both are equally correct and common:

Nos vamos a mudar…
Vamos a mudarnos… With two-verb phrases, the pronoun can go before the conjugated verb or be attached to the infinitive. Meaning and register are the same.

Is mudarse always reflexive? What does mudar mean without se?

mudarse = to move (change residence): Nos vamos a mudar.
mudar (non-reflexive) = to change/replace something: mudar la ropa (change clothes), or animals shedding: la serpiente muda la piel. For “move house,” use the reflexive form.

Why is it a un apartamento and not en un apartamento?

After mudarse, use a to indicate the destination: mudarse a [lugar].
Use de for the place you’re leaving: Nos mudamos de casa a un apartamento.
en would mean “in,” not “to,” so it’s incorrect for the destination.

Does el sábado mean “this Saturday,” or just “on Saturday” in general?

el sábado usually means “on Saturday” (the upcoming one from context).
To be explicit: este sábado = this Saturday; el sábado que viene = next Saturday.
Habitual: los sábados = on Saturdays (every Saturday).

Why is sábado lowercase? And does it need an accent?

Days of the week are not capitalized in Spanish (unless starting a sentence).
sábado always has an accent on the first á. Pronounce it SÁ-ba-do.

Can the time expression move? For example, can I say El sábado nos vamos a mudar…?

Yes. Common options (all fine):

Nos vamos a mudar… el sábado.
El sábado nos vamos a mudar…
Nos vamos el sábado a mudar a… (less common; most natural is to keep a mudar with vamos).

Do I need the article un? Can I say a apartamento?

You need the article. In Spanish, singular countable nouns normally require an article: a un apartamento. Saying a apartamento is ungrammatical here.

Why is the adjective after the noun? Can I say un pequeño apartamento?

Both are correct:

un apartamento pequeño (default, descriptive).
un pequeño apartamento (slightly more subjective/affective or stylistic). In everyday speech, both orders are common; the after-noun position is the neutral default.

Are there regional words for apartamento in Latin America?

Yes:

departamento is very common in much of Latin America (e.g., Mexico, Argentina, Chile).
apartamento is also widely understood and used (e.g., Colombia, Venezuela, Central America).
In Spain you’ll often hear piso.

Could I just use the present: Nos mudamos el sábado? How about Nos mudaremos el sábado?

All three work:

Nos vamos a mudar el sábado: very common “going to” future.
Nos mudamos el sábado: present with near-future meaning; also very natural.
Nos mudaremos el sábado: simple future; correct but less common in casual Latin American speech. Often used for promises or formal contexts.

Is Nos vamos a mudar ever confused with Nos vamos (“We’re leaving”)?

Not in this sentence. Here nos belongs to mudarse, not irse. Nos vamos alone means “we’re leaving,” but Nos vamos a mudar clearly means “we’re going to move (house).” Context and the a + infinitive resolve any ambiguity.

Can I use the progressive: Nos estamos mudando / Estamos mudándonos?

Yes, but it changes the nuance:

Nos estamos mudando = we’re in the process of moving (ongoing).
The original sentence is a plan/scheduled action for Saturday, not necessarily in progress now.

How do I make it negative?

Two correct options:

No nos vamos a mudar el sábado.
No vamos a mudarnos el sábado.

Why is there no contraction like al here?

al is only for a + el (definite article). Here it’s a + un, so no contraction. You would say al apartamento if it were definite: Nos mudamos al apartamento de la esquina.

Are both a’s necessary in Nos vamos a mudar a un…?

Yes:

The first a is part of the future construction ir a + infinitive.
The second a marks the destination after mudarse a. Both must be there.