Ana y mi hermana quieren ir a la biblioteca mañana.

Because the subject is Ana y mi hermana — two people joined by y (and). In Spanish, that makes the subject plural, so the verb must also be plural.

• Ana quiere = Ana wants
• Mi hermana quiere = My sister wants
• Ana y mi hermana quieren = Ana and my sister want

Because querer is a stem-changing verb. In the present tense, the e in the stem changes to ie in most forms.

So:

• yo quiero
• tú quieres
• él/ella quiere
• nosotros queremos
• vosotros queréis
• ellos/ellas quieren

In this sentence, the subject is plural third person, so you use quieren.

Because after querer (to want), Spanish normally uses an infinitive when saying what someone wants to do.

So the pattern is:

querer + infinitive

Examples:

• Quiero comer = I want to eat
• Queremos salir = We want to go out
• Ana y mi hermana quieren ir = Ana and my sister want to go

You do not conjugate the second verb here, so it stays as ir, not van.

Because after a conjugated verb like quieren, the next verb is usually left in the infinitive.

• Correct: quieren ir
• Incorrect: quieren van

This is similar to English:

• They want to go not
• They want they go

Because ir usually takes a when you say where someone is going:

• ir a casa
• ir al supermercado
• ir a la biblioteca

Here, biblioteca is a feminine noun, so it uses la:

• la biblioteca

That gives:

• a + la biblioteca = a la biblioteca

In Spanish, only these two combinations contract:

• a + el = al
• de + el = del

There is no contraction with la, so:

• a la biblioteca stays a la biblioteca

Compare:

• Voy al parque
• Voy a la biblioteca

In standard modern Spanish, possessive adjectives like mi, tu, su, nuestro, etc. normally replace the article.

So you say:

• mi hermana not
• la mi hermana

This is different from English only in structure, not meaning. Mi hermana simply means my sister.

Because mi does not change for gender. It only changes for number:

• mi hermano
• mi hermana
• mis hermanos
• mis hermanas

So mi works with both hermano and hermana.

Because Spanish often puts time expressions like mañana, hoy, or esta tarde at the end, but the position is flexible.

All of these are possible:

• Ana y mi hermana quieren ir a la biblioteca mañana.
• Mañana, Ana y mi hermana quieren ir a la biblioteca.
• Ana y mi hermana mañana quieren ir a la biblioteca.

The version with mañana at the end sounds very natural.

It means they want to go tomorrow. The main verb is quieren (want), so the sentence expresses desire or intention, not a definite future action by itself.

If you wanted to say they are definitely going tomorrow, you might say:

• Ana y mi hermana van a la biblioteca mañana.

If you wanted to emphasize a future plan, you could also say:

• Ana y mi hermana van a ir a la biblioteca mañana.

Because personal names normally do not take an article in standard Spanish.

So:

• Ana not usually
• la Ana

In some regional or informal varieties, people may sometimes use an article with names, but for standard learning purposes, just use Ana without one.

Yes, very often Spanish omits subject pronouns like ellos or ellas because the verb ending already gives information.

For example:

• Quieren ir a la biblioteca mañana.

This could mean They want to go to the library tomorrow, depending on context.

But in your sentence, Ana y mi hermana is included because the speaker wants to specify exactly who they are.

Spanish normally uses y for and.

It changes to e only before words that begin with an i sound:

• padre e hijo
• España e Italia

But here the next word is mi, which begins with an m sound, so it stays:

• Ana y mi hermana

Yes, biblioteca is a feminine noun, so it takes feminine articles and adjectives:

• la biblioteca
• una biblioteca
• la biblioteca grande

That is why the sentence has a la biblioteca.

Yes, Spanish word order is more flexible than English, although some orders sound more natural than others.

Natural versions include:

• Ana y mi hermana quieren ir a la biblioteca mañana.
• Mañana Ana y mi hermana quieren ir a la biblioteca.
• A la biblioteca quieren ir Ana y mi hermana mañana. — more marked or emphatic

The original sentence is the most neutral and natural for everyday use.

It would change the meaning.

• quieren ir = they want to go
• van a ir = they are going to go

So:

• Ana y mi hermana quieren ir a la biblioteca mañana.
  = They want to go tomorrow.

• Ana y mi hermana van a ir a la biblioteca mañana.
  = They are going to go tomorrow.

That is an important difference: desire/intention vs planned future.