Eu vou fazer o pedido depois de olhar o cardápio.

That’s true: in Portuguese, the subject pronoun is often omitted when the verb already makes the subject clear.

So this sentence could also be:

Vou fazer o pedido depois de olhar o cardápio.

Both are correct.

Including eu can:

• add emphasis
• make the sentence clearer in context
• sound a little more explicit

In everyday Brazilian Portuguese, dropping eu is very common if there is no risk of confusion.

Yes, literally vou fazer means I am going to do/make, but in this sentence it works like a very natural future expression:

• vou fazer = I’m going to do / I’m going to make
• vou fazer o pedido = I’m going to place the order

This is one of the most common ways to talk about the future in Brazilian Portuguese:

• vou comer = I’m going to eat
• vou sair = I’m going to leave
• vou ligar = I’m going to call

It is often more common in speech than the simple future form.

Both are possible, but they are not exactly the same structure.

• fazer o pedido = literally to make/place the order
• pedir = to order / to ask for

So you could also say:

Eu vou pedir depois de olhar o cardápio.

That means something like I’m going to order after looking at the menu.

The version with fazer o pedido sounds a bit more like place the order, while pedir is shorter and very common in conversation.

Pedido is a noun. In this sentence, it means order.

It comes from the verb pedir, which means to ask for or to order.

Compare:

• pedir = to order / to ask for
• o pedido = the order

Examples:

• Vou pedir agora. = I’m going to order now.
• O pedido chegou. = The order arrived.

So in your sentence, fazer o pedido is a noun phrase built around pedido.

Because depois de is the normal structure when it is followed by an infinitive verb.

So:

• depois de olhar = after looking
• depois de comer = after eating
• depois de sair = after leaving

This is a very important pattern in Portuguese:

• antes de sair = before leaving
• depois de estudar = after studying

If you use a full clause with a subject and conjugated verb, then you normally use depois que instead:

• depois que eu olhar o cardápio = after I look at the menu

So both are possible, but they have different grammar:

• depois de + infinitive
• depois que + conjugated verb

In Portuguese, olhar can be used both ways, but the meaning can shift a little.

• olhar o cardápio = to look at the menu, to examine/read it
• olhar para o cardápio = to look toward/at the menu

When you are talking about reading or checking a menu, olhar o cardápio sounds very natural.

A rough distinction is:

• olhar + noun often feels more direct
• olhar para + noun emphasizes the direction of your gaze

In a restaurant context, olhar o cardápio is completely normal.

Those are the definite articles the.

• o pedido = the order
• o cardápio = the menu

Portuguese uses articles very often, sometimes more often than English does. In this sentence, they sound natural because the speaker is talking about:

• the specific order they are about to make
• the specific menu in front of them

Without the articles, the sentence would sound unnatural in this context.

Yes, but cardápio is usually the more standard and common word for a restaurant menu in Brazilian Portuguese.

• cardápio = menu
• menu also exists, but depending on context it can sound less traditional or can refer to other kinds of menus, like a computer menu

In restaurants in Brazil, cardápio is a very safe and natural choice.

They are similar in meaning, but not identical in tone.

• vou fazer o pedido = I’m going to place the order
• farei o pedido = I will place the order

In Brazilian Portuguese, vou fazer is much more common in everyday speech.

Farei is correct, but it often sounds:

• more formal
• more written
• sometimes less conversational

So in normal spoken Brazilian Portuguese, vou fazer o pedido is the more natural choice.

Yes. Both are possible, but they are slightly different in nuance.

• olhar o cardápio = to look at the menu
• ver o cardápio = to see/check the menu

In practice, both can work in a restaurant context.
However, olhar o cardápio often feels a little more natural because it suggests actually taking time to examine the menu before ordering.

The original word order is very natural:

Eu vou fazer o pedido depois de olhar o cardápio.

You can move parts of the sentence around, but the tone changes slightly.

For example:

Depois de olhar o cardápio, eu vou fazer o pedido.

This is also correct and natural. It puts more focus on after looking at the menu.

Portuguese allows this kind of flexibility, but the original version is probably the most neutral and conversational.

The stress is on dá:

car-DÁ-pio

A rough English-friendly approximation is:

kar-DAH-pyoo

A few pronunciation notes:

• car sounds like car without a strong English r
• dá is the stressed syllable
• pio is said together, roughly like pyoo

The written accent in cardápio shows where the stress goes.