Después de la escuela, estudio en la biblioteca y luego vuelvo a casa.

Because después de is the normal pattern when after is followed by a noun.

• después de la escuela = after school
• después de la clase = after class

If what follows is a full clause, Spanish usually uses después de que:

• Después de que termino, vuelvo a casa = After I finish, I go back home

So in this sentence, de is needed because la escuela is a noun phrase.

Spanish often uses the definite article where English does not.

So:

• después de la escuela
• en la biblioteca
• voy al trabajo
• estoy en la oficina

This is very normal in Spanish. You should not try to match article use directly with English.

It is correct and fully understandable, but in Spain the most natural word depends on the type of school.

Common choices are:

• colegio = primary school, sometimes school in general
• instituto = secondary school
• escuela = a more general word for school, or a specific kind of school such as escuela de idiomas, escuela de música

So the sentence is grammatical, but in everyday Spain Spanish, many people might say:

• Después del colegio...
• Después del instituto...

depending on the student’s age.

Spanish often leaves out subject pronouns because the verb ending already shows who the subject is.

• estudio = I study
• vuelvo = I return / I go back

So yo is not necessary here.

You would include yo only for emphasis, contrast, or clarity:

• Yo estudio en la biblioteca, pero mi hermano estudia en casa.

It can mean either, depending on context.

In Spanish, the simple present often covers both:

• I study
• I am studying

In this sentence, it most naturally sounds habitual:

• After school, I study in the library and then I go home.

If you specifically want to stress that the action is happening right now, Spanish often uses estar + gerund:

• Estoy estudiando en la biblioteca. = I am studying in the library.

Because en shows location, while a usually shows movement toward a destination.

• estudio en la biblioteca = I study in the library
• voy a la biblioteca = I go to the library

In the sentence, the person is already studying there, so en is the correct preposition.

Here luego means then or afterwards.

• Después de la escuela sets the starting point: after school
• luego adds the next step in the sequence: then I go home

In many contexts, luego and después are very close in meaning:

• Luego vuelvo a casa
• Después vuelvo a casa

Both are fine. In this sentence, luego sounds very natural.

Because volver is a stem-changing verb in the present tense: o → ue.

Its present forms are:

• vuelvo
• vuelves
• vuelve
• volvemos
• volvéis
• vuelven

So vuelvo is the correct I form.

This kind of stem change is very common in Spanish.

A casa is the normal idiomatic way to say home.

• volver a casa = to go back home
• llegar a casa = to arrive home
• estar en casa = to be at home

If you say a la casa, it usually sounds more like to the house/building, not home in the general personal sense.

So:

• Vuelvo a casa = I go back home
• Vuelvo a la casa = I return to the house

Not strictly.

Both of these are acceptable:

• Después de la escuela estudio en la biblioteca y luego vuelvo a casa.
• Después de la escuela, estudio en la biblioteca y luego vuelvo a casa.

The comma marks a small pause after the introductory phrase. In short introductory phrases, Spanish often leaves the comma out, but some writers include it for rhythm or clarity.

Yes. Spanish word order is more flexible than English, as long as the meaning stays clear.

For example:

• Después de la escuela, estudio en la biblioteca y luego vuelvo a casa.
• Estudio en la biblioteca después de la escuela y luego vuelvo a casa.
• Luego vuelvo a casa, después de estudiar en la biblioteca.

The original order is natural because it presents the events in time order.

The accent mark shows where the stress goes: des-pués.

Without the accent, Spanish stress rules would suggest a different pronunciation. Since después ends in s, a word without an accent would normally be stressed on the second-to-last syllable. The accent tells you that the stress falls on the last syllable instead.

So the pronunciation is roughly:

• des-PWES

Yes. Regreso a casa and vuelvo a casa are both correct.

However, volver a casa is usually more common and more everyday in this kind of sentence.

• vuelvo a casa = very common, natural
• regreso a casa = also correct, sometimes a bit more formal or literary depending on context

So vuelvo is a very good choice here.