Después de la escuela, caminamos por el puente hacia la biblioteca.

Because después normally needs de when it is followed by a noun, pronoun, or infinitive.

• después de la escuela = after school
• después de comer = after eating

You can use después by itself when it means afterwards or later:

• Fuimos a casa y después cenamos. = We went home and afterwards we had dinner.

In Spanish, de + el becomes del, but only with the masculine article el.

• de + el parque = del parque
• de + la escuela = de la escuela

Since escuela is feminine, it takes la, so there is no contraction.

It is correct and fully understandable, but in Spain the most everyday word often depends on the type of school:

• colegio = primary/elementary school, and often a general everyday word for school
• instituto = secondary school
• escuela = a more general term, or used in certain names such as escuela de idiomas

So the sentence is fine, but in many everyday situations in Spain you might also hear Después del colegio... or Después del instituto..., depending on the context.

Caminamos is the first person plural form of caminar: we walk or we walked.

This form can mean either:

• present: we walk
• preterite: we walked

Spanish often relies on context to tell you which one it is. In a sentence like this, if the overall meaning is past, then caminamos means we walked.

Caminar specifically means to walk.

• caminamos = we walked / we walk
• fuimos = we went
• andamos = we walk / we go around / we get about

So caminamos tells you that the movement was on foot.
Fuimos would be more general and would not say how you got there.
Andar can also mean to walk, especially in Spain, but it has a wider range of meanings, so caminar is a bit more specific here.

Here por shows the route or path taken.

So por el puente means something like:

• along the bridge
• across the bridge
• by way of the bridge

It tells you which way you went. Spanish often uses por for movement through or along a place:

• Pasamos por el parque. = We went through the park.
• Entraron por la puerta principal. = They came in through the main door.

If you wanted to focus more directly on the act of crossing, Spanish might also say cruzamos el puente.

Because they express two different ideas:

• por el puente = the route taken
• hacia la biblioteca = the direction of movement

So the sentence gives both pieces of information:

• where we went by
• where we were heading

This is very natural in Spanish.

Hacia means toward or in the direction of, while a usually means to.

So:

• hacia la biblioteca = toward the library
• a la biblioteca = to the library

Hacia does not always guarantee arrival; it mainly gives the direction.
A la biblioteca sounds more like the library is the definite destination.

Spanish uses definite articles more often than English.

In English, we often say after school with no article. In Spanish, the natural form is usually:

• después de la escuela
• después del colegio

Likewise, la biblioteca refers to a specific library understood from the context. Spanish generally prefers the article with common places like this.

It is common and perfectly correct because Después de la escuela is an introductory phrase.

• Después de la escuela, caminamos...
• Después de la escuela caminamos...

Both are possible. The comma simply marks a small pause and can make the sentence easier to read.

Because without the accent, Spanish stress rules would make you pronounce it incorrectly.

Words ending in a vowel, n, or s are normally stressed on the second-to-last syllable. But después is stressed on the last syllable, so it needs a written accent to show that:

• de-spués

The accent tells you where the stress goes.