Después de clase, suelo saludar al librero de la esquina antes de volver a casa.

Después de clase is the usual way to mean after class in a general, routine sense.

• después de clase = after class / after school, as part of a normal schedule
• después de la clase = after the class, referring to one specific class

So in this sentence, después de clase sounds natural because the speaker is talking about a habit, not one particular lesson.

Suelo comes from the verb soler, which means to usually do something or to tend to do something.

So:

• suelo saludar = I usually greet / I tend to greet

It is followed by an infinitive:

• suelo + infinitive
• suelo saludar
• sueles estudiar
• solemos comer tarde

This is a very common way in Spanish to talk about habits.

Both are possible, but they feel slightly different:

• suelo saludar = I usually greet / I tend to greet
• normalmente saludo = normally I greet / I normally greet

Soler + infinitive is very natural when talking about repeated habits. It often sounds smoother and more idiomatic than using an adverb like normalmente.

So suelo saludar is a very natural Spanish way to express a routine.

Al is a contraction of a + el:

• a el → al

It appears here because saludar a alguien takes the preposition a before a person:

• saludar al librero
• saludar a mi amiga
• saludar al profesor

So al librero means to the bookseller or the bookseller as the person being greeted.

This is because of the personal a, a very common feature in Spanish.

When the direct object is a specific person, Spanish usually puts a before it:

• Veo a María
• Conozco al vecino
• Saludo al librero

Even though English does not do this, Spanish normally does when a person is the object of the verb.

Here, librero means bookseller or bookshop owner/bookseller working in a bookshop.

This can be confusing because of related words:

• librería = bookshop / bookstore
• biblioteca = library

So librero is the person connected with bookselling, not a library worker.

De la esquina means on the corner or literally of the corner.

In Spanish, this structure is often used to identify someone or something by location:

• el bar de la esquina = the bar on the corner
• la tienda de la esquina = the shop on the corner
• el librero de la esquina = the bookseller on the corner

So it tells you which bookseller the speaker means.

After antes de, Spanish uses an infinitive when the subject stays the same:

• antes de volver a casa = before returning home / before going back home

This is the normal pattern:

• antes de salir
• antes de comer
• antes de estudiar

So antes de + infinitive is the structure you need here.

In Spanish, casa often appears without an article when it means home:

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

But la casa means the house, the physical building:

• volví a la casa de mi abuela = I went back to my grandmother’s house

So volver a casa is the natural choice for go back home.

Yes. Volver means to return, to go back, or to come back, depending on context.

In this sentence:

• volver a casa = to go back home

It is one of the most common verbs for returning somewhere. Another possible verb is regresar, but volver is extremely common and natural in everyday Spanish.

The present tense is used because the sentence describes a habitual action:

• Después de clase, suelo saludar... = After class, I usually greet...

Spanish often uses the present tense for routines and repeated actions, just like English does:

• Después del trabajo, ceno temprano.
• Los domingos, solemos visitar a mis abuelos.

So the present here does not mean only right now; it can also describe habits.

Yes, Spanish word order is flexible, but the original sentence is very natural.

Original:

• Después de clase, suelo saludar al librero de la esquina antes de volver a casa.

You could also say:

• Suelo saludar al librero de la esquina después de clase, antes de volver a casa.

Both are understandable, but the original puts después de clase first to set the scene right away. That is very common in Spanish.

It is very natural and helpful, because Después de clase is an introductory time phrase.

• Después de clase, suelo saludar...

The comma makes the sentence easier to read. In short sentences, people sometimes omit it, but using it here is perfectly standard and clear.