Meu neto quer aprender a nadar neste verão.

Because neto is a masculine singular noun, so the possessive must match it: meu.

• meu neto = my grandson
• minha neta = my granddaughter

In Portuguese, the possessive agrees with the thing being possessed, not with the speaker.

Quer is the 3rd person singular present tense of querer, meaning to want.

It matches meu neto, which is he:

• eu quero = I want
• ele/ela quer = he/she wants
• nós queremos = we want
• eles/elas querem = they want

So Meu neto quer... means My grandson wants...

Portuguese often omits subject pronouns when the subject is already clear.

So:

• Meu neto quer aprender...
• Ele quer aprender...

Both are possible, but once meu neto is stated, adding ele would usually be unnecessary.

This is very normal in Portuguese and happens much more often than in English.

After aprender, Portuguese commonly uses a + infinitive to mean to learn to do something.

So:

• aprender a nadar = to learn to swim
• aprender a dirigir = to learn to drive
• aprender a cozinhar = to learn to cook

Here, the a is not translated word-for-word. It is just part of the normal Portuguese pattern.

The safest and most standard choice is aprender a nadar.

You may hear aprender nadar in some speech, but aprender a + infinitive is the form learners should use.

So if you want to sound natural and correct, use:

Meu neto quer aprender a nadar.

There is an important difference:

• quer aprender a nadar = wants to learn to swim
• quer nadar = wants to swim

The first suggests he does not yet know how, or at least wants instruction or practice.
The second only says he wants to do the activity.

Neste is a contraction of:

em + este = neste

So neste verão literally means in this summer, which English expresses as this summer.

This kind of contraction is very common in Portuguese:

• no = em + o
• na = em + a
• neste = em + este
• nesse = em + esse

In careful grammar:

• neste verão = this summer
• nesse verão = that summer or that summer you mentioned

However, in everyday Brazilian Portuguese, people often use esse/nesse where grammar books might prefer este/neste.

So neste verão is fully correct and clear, and nesse verão may also be heard depending on context and speaking style.

Because no verão and neste verão are not exactly the same.

• no verão = in the summer / during summer in a more general sense
• neste verão = this summer, referring to a specific summer

So:

• Meu neto quer aprender a nadar no verão = My grandson wants to learn to swim in summer / during the summer
• Meu neto quer aprender a nadar neste verão = My grandson wants to learn to swim this summer

Yes. Both are possible.

• meu neto
• o meu neto

In Brazilian Portuguese, meu neto without the article is very common and natural.
Adding the article can sound a little more specific, emphatic, or regional, but it is not wrong.

For a learner, meu neto is a very good default choice.

The ã shows a nasal vowel. In verão, the ending -ão is one of the most common nasal endings in Portuguese.

A rough English-style approximation of verão is veh-ROWN, but that is only approximate. The real Portuguese sound is more nasal and not exactly like English.

A few helpful points:

• stress falls on the last syllable: ve-RÃO
• ão is nasal
• the accent mark also helps show the stressed syllable

This -ão ending appears in many common words, such as:

• não
• pão
• coração

Yes. Portuguese allows some flexibility.

For example:

• Meu neto quer aprender a nadar neste verão.
• Neste verão, meu neto quer aprender a nadar.

Both are natural.
Putting Neste verão first gives a little more emphasis to the time expression.

Yes. It sounds completely natural and correct in Brazilian Portuguese.

It is a normal, everyday sentence, and the grammar is standard:

• Meu neto = subject
• quer = main verb
• aprender a nadar = infinitive phrase
• neste verão = time expression

So this is a very good model sentence for learners.