Eu preciso comprar alface, cenoura e manteiga no supermercado.

Yes. In Brazilian Portuguese, subject pronouns are often optional because the verb form already shows the person.

So both of these are natural:

• Eu preciso comprar alface, cenoura e manteiga no supermercado.
• Preciso comprar alface, cenoura e manteiga no supermercado.

Including eu can add clarity or emphasis, but it is not required here.

With a second verb in the infinitive, Brazilian Portuguese most commonly uses:

• precisar + infinitive

So:

• preciso comprar
• preciso estudar
• preciso sair

When precisar is followed by a noun, de is very common:

• Preciso de manteiga.
• Preciso de dinheiro.

So a helpful pattern is:

• precisar + verb
• precisar de + noun

Because preciso is the conjugated verb, and comprar is the action that is needed.

This is like English I need to buy:

• preciso = I need
• comprar = to buy

After verbs like precisar, querer, poder, and dever, the next verb usually stays in the infinitive.

Because this sounds like a general shopping list, not specific individual items already identified.

In Portuguese, when talking about things you need to buy in general, it is very common to say:

• comprar alface
• comprar cenoura
• comprar manteiga

If you wanted to be more specific, articles could appear:

• comprar a manteiga que eu gosto
• comprar a alface da feira

But in a simple list, leaving the articles out is very normal.

Because no is a contraction:

• em + o = no

And supermercado is a masculine noun:

• o supermercado

So:

• no supermercado = in/at the supermarket

Other common contractions work the same way:

• na = em + a
• nos = em + os
• nas = em + as

Alface is feminine:

• a alface

This can surprise learners because the word does not end in -a, but it is still feminine.

Examples:

• a alface fresca
• a alface está cara

Portuguese often uses the singular in shopping-list style language to refer to a food item in general.

So cenoura can mean the product/item category, not necessarily exactly one carrot.

That said, cenouras is also possible if you are thinking of individual carrots. Compare:

• Preciso comprar cenoura. → general shopping-list style
• Preciso comprar cenouras. → clearly multiple carrots

Both can be natural, depending on how specific you want to be.

No. Comprar normally takes a direct object, so the items come right after the verb with no extra preposition.

• comprar alface
• comprar cenoura
• comprar manteiga

This is similar to English buy lettuce, buy carrots, buy butter.

Yes, often you can. Both are common, but they are not always exactly the same.

• supermercado usually means a supermarket
• mercado can be more general and may mean a market or grocery store, depending on context

In everyday speech, many Brazilians use both naturally, depending on the place they mean.

A simple approximate pronunciation is:

eh-ooh preh-SEE-zoo kohm-PRAHR aw-FAH-see, seh-noh-RAH ee mahn-TAY-gah noo soo-pehr-mehr-KAH-doo

A few useful notes:

• preciso has stress on -ci-
• cenoura has stress on -nou-
• manteiga sounds roughly like man-TAY-ga
• supermercado has stress on -ca-
• e before a consonant is usually pronounced like ee

Normally, no. In Portuguese, as in standard English style, items in a simple list are separated by commas, but there is usually no comma before e.

So this is the normal punctuation:

• alface, cenoura e manteiga

That final comma before e is generally not used in ordinary Portuguese writing.