Learn Hardcore Portuguese (Portugal): Eu entrei num clube de leitura para falar mais sobre livros. - I joined a book club to talk more about books.

Entrei is the past tense (pretérito perfeito) of entrar and literally means I entered.
In European Portuguese, entrar num/para um clube can also mean to join a club, not just physically walk into the building.

Eu entrei num clube de leitura… = I joined a reading club… (or I went into a reading club, depending on context).
Eu fui a um clube de leitura… = I went to a reading club (I went there; it doesn’t clearly say I became a member).
Eu juntei-me a um clube de leitura… = I joined a reading club (very clear about membership).

Your sentence is natural in European Portuguese as a way to say you joined, especially with the purpose clause para falar mais sobre livros.

What exactly is num? Is it different from no or em um?

Num is a contraction:

em + um = num → in/into a
em + o = no → in/into the

So:

num clube de leitura = in/into a reading club
no clube de leitura = in/into the reading club (a specific one we both know)

Em um is grammatically correct and means the same as num, but in European Portuguese people almost always use the contraction num in speech and normal writing. Using em um sounds more formal, written, or Brazilian-influenced.

Can I drop Eu and just say Entrei num clube de leitura…?

Yes. In Portuguese, subject pronouns are often omitted because the verb ending already shows who the subject is.

Eu entrei num clube de leitura… – grammatically fine, slightly more emphasis on I.
Entrei num clube de leitura… – very natural, neutral statement.

You typically keep Eu if you want to:

contrast with someone else (Eu entrei, mas ela não)
or give special emphasis to yourself.

Why do we say entrar em / entrar num and not just entrar with a direct object?

In Portuguese, entrar normally takes the preposition em (or its contractions no/na/nos/nas, num/numa/numas):

Entrar em casa
Entrar na sala
Entrar num clube de leitura

You cannot say ✗ entrar um clube. That sounds wrong to a native speaker.

There is also entrar para when talking about joining an organization:

Entrei para um clube de leitura.

Both entrar em and entrar para can express joining, depending on context.

What does clube de leitura literally mean, and is it the same as clube do livro?

Literally, clube de leitura is club of reading, i.e. a reading club / book club.

You might also see:

clube do livro – literally club of the book.

Nuance (not a strict rule, but a tendency):

clube de leitura – focuses on the activity of reading and discussing.
clube do livro – often suggests a group around a particular book each time (like a book-of-the-month club).

In practice, both can be understood as a book club, and clube de leitura is very natural in Portugal.

Why is it para falar and not por falar or a falar?

Para + infinitive expresses purpose / intention:

Entrei num clube de leitura para falar mais sobre livros.
→ I joined in order to talk more about books.

Other options mean something different:

por falar – usually because of speaking / by speaking (reason or means), not purpose.
a falar – often used for a continuous action (estava a falar = was speaking) or in a few set expressions; not natural here for purpose.

So for in order to do X, use para + infinitive: para falar, para estudar, para aprender, etc.

Why is it the infinitive falar and not something like falarmos or para eu falar?

After para you normally use the infinitive to express purpose:

para falar mais sobre livros = to talk more about books.

You could see two related forms:

Personal infinitive: para falarmos
- Used when the subject is we:
  Entrei num clube de leitura para falarmos mais sobre livros.
  → I joined so that we can talk more about books.
Explicit subject: para eu falar
- Stresses that I am the one who will speak:
  Entrei num clube de leitura para eu falar mais sobre livros.

Your original sentence (with simple falar) is the most neutral and typical way to express your purpose.

What exactly does mais mean here, and can I move it or sobre livros to other positions?

Here mais means more, and in this context it usually means:

more often or
to a greater extent / in more depth.

So para falar mais sobre livros = to talk more about books (either more frequently or more/longer, depending on context).

Word order changes the meaning:

para falar mais sobre livros – talk more about the topic of books.
para falar sobre mais livros – talk about more books (a greater number / variety of books).

Something like ✗ para falar sobre livros mais is unnatural in Portuguese.

Why is it livros (plural)? Could I say livro, os livros, or o livro instead?

Livros (plural, no article) means books in general:

falar mais sobre livros = talk more about books as a general topic.

Alternatives:

sobre o livro – about the book (one specific book).
sobre os livros – about the books (a specific group of books we know about).
sobre livro (singular, no article) is possible but unusual here; it sounds more like “as a topic: book”, and natives almost always say livros when they mean the general theme of books.

Why is entrei in this past tense and not something like entrava?

Entrei is pretérito perfeito, used for a finished, one-time action in the past:

Eu entrei num clube de leitura… = I joined (at a specific point in time).

Entrava is pretérito imperfeito, usually used for:

repeated / habitual actions in the past, or
background descriptions.

For example:

Quando era mais novo, entrava em muitos clubes só para conhecer gente.
→ When I was younger, I used to join many clubs just to meet people.

If you want to talk about a past habit related to this sentence, you’d more likely change the verb entirely:

Eu ia a um clube de leitura para falar mais sobre livros.
→ I used to go to a reading club to talk more about books.

Is clube masculine or feminine, and how does that affect the article?

Clube is masculine in Portuguese.

That’s why you see:

um clube (a club)
o clube (the club)
num clube = em + um clube
no clube = em + o clube

If it were feminine, you’d use uma / a / numa / na, but you don’t with clube.
Gender is mostly lexical in Portuguese, so you usually just have to learn each noun’s gender (dictionaries mark m. for masculine and f. for feminine).

Are there any important differences between European and Brazilian Portuguese for this sentence?

The sentence works in both varieties, but there are tendencies:

European Portuguese (Portugal)
Eu entrei num clube de leitura para falar mais sobre livros.
– very natural, especially with num.
Brazilian Portuguese
Brazilians might say:
- Eu entrei em um clube de leitura… (less contracted), or
- Eu entrei para um clube de leitura…, or
- More commonly for ongoing membership: Eu entrei em / entrei para / participo de um clube de leitura…

Prosody and pronunciation also differ: European Portuguese reduces many unstressed vowels more strongly than Brazilian Portuguese.

How do you pronounce this sentence in European Portuguese?

In a careful, standard European pronunciation, roughly:

Eu entrei num clube de leitura para falar mais sobre livros.
→ [ew ẽˈtɾɐj nũ ˈklub(ɨ) dɨ lɐjˈtuɾɐ ˈpaɾɐ fɐˈlaɾ majʃ ˈsobɾɨ ˈlivɾuʃ]

Some notes:

Eu → [ew]
entrei → stress on -trei: [ẽˈtɾɐj]
num → nasal vowel: [nũ]
clube → [ˈklub(ɨ)] (final e is very reduced)
de → [dɨ]
leitura → [lɐjˈtuɾɐ]
para → often [ˈpaɾɐ] or reduced to [pɾɐ] in fast speech
mais → [majʃ]
sobre → [ˈsobɾɨ]
livros → [ˈlivɾuʃ]

Actual everyday speech will be a bit faster and with even more vowel reduction.