Depois eu vou descansar em casa.

You don’t need eu here. Both are correct:

• Depois eu vou descansar em casa. – grammatically correct, with explicit eu.
• Depois vou descansar em casa. – also correct, and more typical in everyday European Portuguese.

Portuguese verb endings already show the subject (vou = I go / I’m going), so the pronoun is often dropped unless you want:

• emphasis or contrast: Eu vou descansar em casa, tu vais sair.
• extra clarity in a complex context.

In your sentence, most speakers from Portugal would naturally say Depois vou descansar em casa.

Vou descansar is ir + infinitive, a very common way to talk about the (near) future in Portuguese. It feels like saying “I’m going to rest” in English.

• Eu descanso em casa. – usually means a habit: I (usually) rest at home.
• Eu vou descansar em casa. – means a future action: I’m going to rest at home (later / after this).

So vou descansar is chosen here because we’re talking about something that will happen after something else, not a general habit.

Grammatically yes, but it sounds very formal or literary in modern spoken European Portuguese:

• Depois descansarei em casa. – correct, but sounds like written style, speeches, or very deliberate/solemn talk.
• Depois vou descansar em casa. – this is what people normally say in everyday conversation.

In practice, the ir + infinitive form (vou descansar) has largely replaced the simple future (descansarei) in spoken Portuguese.

In this sentence, depois at the start means “afterwards / then / later” in a general time sense, not “after [something] specific”.

• Depois eu vou descansar em casa.
  → After that / afterwards / later on, I’m going to rest at home.

When depois is followed by de, it usually means “after [something specific]”:

• Depois de jantar, vou descansar em casa. – After dinner, I’m going to rest at home.
• Depois de trabalhar, vou descansar em casa. – After working, I’m going to rest at home.

So:

• depois alone = later / afterwards
• depois de
  • noun / verb = after [X]

Yes, but the most neutral options are:

• Depois vou descansar em casa.
• Depois eu vou descansar em casa.

Other positions are possible and still correct, but they change emphasis slightly:

• Eu depois vou descansar em casa. – emphasises eu a bit first, then adds depois; sounds a bit more marked or “spoken-style”.
• Eu vou descansar em casa depois. – sounds like “I’ll rest at home later (not now)”; depois feels more tightly linked to the verb phrase.

For a learner, stick to Depois vou descansar em casa as your “default” pattern.

Em casa is a fixed, very common expression meaning “at home” (not “in the house” in a general sense).

• em casa = at home (your home, or “home” in general)
• na casa = in/at the house (some specific house, with the article)

Note:

• em + a must contract to na, so em a casa is not used; it automatically becomes na casa.
• But with casa meaning “home”, Portuguese does not use the article, so we say em casa, not na casa, when we mean “at home”.

Examples:

• Estou em casa. – I’m at home.
• Estou na casa dos meus avós. – I’m at my grandparents’ house.

• em casa → location: at home
• para casa → direction/movement: (going) home / to home

Compare:

• Depois vou descansar em casa. – Later I’ll be (resting) at home.
• Depois vou para casa descansar. – Later I’m going home to rest.

So:

• Use em casa when you are already there.
• Use para casa when you are going there.

In normal modern use, descansar is not reflexive when it means “to rest”:

• Vou descansar. – I’m going to rest.
• Ele precisa de descansar. – He needs to rest.

You might occasionally see descansar-se, but it sounds old-fashioned or has a slightly different nuance (more like “take it easy, calm down” in some contexts). For everyday “rest”, just use descansar without -se.

Yes. Depois eu vou descansar is a complete, natural sentence.

• Depois eu vou descansar. – Later I’m going to rest (no place specified).
• Depois eu vou descansar em casa. – adds where you’ll rest.

So em casa is optional and only specifies the location.

Both are correct, but in European Portuguese the more neutral, everyday version is:

• Depois vou descansar em casa.

Dropping eu is very common because the verb form vou already shows the subject.
Depois eu vou descansar em casa is still fine, and might be used:

• if you want to emphasise I (as opposed to someone else), or
• in clearer, slower speech to beginners.

In (standard) European Portuguese, a careful pronunciation would be roughly:

• Depois – [dɨˈpɔjʃ] (final s like English sh)
• eu – [ew]
• vou – [vo] (like “voh”, a single syllable)
• descansar – [dɨʃkɐ̃ˈsaɾ]
• em – [ẽ] (nasalised “eh”)
• casa – [ˈkazɐ]

All together (in IPA, fairly careful speech):
[dɨˈpɔjʃ ew vo dɨʃkɐ̃ˈsaɾ ẽ ˈkazɐ]

In faster everyday speech:

• vowels like the e in depois and descansar get very reduced (like a weak “uh”),
• depois eu may sound almost like [d’pɔjʒew], with the sounds running together.

You don’t need to replicate native speed at first; aim for the syllables and stress:

De-POIS eu VOU des-can-SAR em CA-sa.

Yes, that’s possible:

• Eu vou descansar em casa depois.

This usually feels like:

• “I’ll rest at home later (not now / not earlier).”

Putting depois at the start (Depois vou descansar em casa) sounds slightly more like a step in a sequence of actions, like “First I’ll do X, then I’ll rest at home.”
At the end, depois more strongly emphasises the time (“later on”) rather than the sequence in a narrative.

Both are correct; for a simple, neutral sentence, Depois vou descansar em casa is a very safe choice.