Després de la classe, vull mirar el correu.

Because després normally needs the preposition de before a noun or pronoun.

• després de la classe = after class
• després d’això = after that

So de is required here.

In Catalan, de + el = del, but de + la stays de la.

So:

• de + el correu → del correu
• de + la classe → de la classe

That is why de la classe is correct.

Vull means I want. It comes from the verb voler, which means to want.

This is the 1st person singular form:

• jo vull = I want

A few present-tense forms of voler are:

• jo vull = I want
• tu vols = you want
• ell/ella vol = he/she wants
• nosaltres volem = we want

Yes, mirar literally means to look, but it is also commonly used in Catalan in the sense of to check, to look at, or to have a look at something.

So mirar el correu can naturally mean:

• to check the mail
• to check email
• to look at the mail

In everyday language, this is very normal.

It can mean either, depending on context.

• correu = mail/post
• correu electrònic = email

In modern everyday use, mirar el correu often means to check email, but the fuller phrase mirar el correu electrònic is more explicit if you want to avoid ambiguity.

Catalan often uses the definite article where English does not.

So Catalan says:

• mirar el correu

where English may say:

• check email
• check the mail

This is normal Catalan usage. The article does not necessarily mean a specific single email or letter; it can refer to the general category.

For the same general reason: Catalan often uses articles more broadly than English.

So després de la classe literally looks like after the class, but in context it can simply mean after class.

This is very natural in Catalan.

Yes. That is also correct.

Both are natural:

• Després de la classe, vull mirar el correu.
• Vull mirar el correu després de la classe.

The first version puts more focus on the time expression after class.
The second starts with the main action I want to check the mail/email.

Because that opening time phrase is being separated from the main clause.

• Després de la classe, vull mirar el correu.

The comma helps mark after class as an introductory element. In short sentences, punctuation can vary a bit, but this comma is very natural and clear.

In standard Catalan, ll represents a palatal sound, traditionally like the lli sound in million for many English speakers, though actual pronunciation varies by dialect.

So vull is approximately:

• booly / buyll as a rough guide for English speakers

But this is only approximate. The important point is that ll is not a normal English l sound.

Also, stress in the sentence falls naturally like this:

• desPRÉS
• CLASse
• VULL
• miRAR
• corREU

Yes, but the nuance changes slightly.

• mirar el correu = check/look at the mail/email
• revisar el correu = review/check the mail/email more carefully
• comprovar el correu = check/verify the mail/email, a bit less common in this everyday context

For casual everyday speech, mirar el correu is very idiomatic.

Yes, it is a very normal way to say after class.

Some alternatives are possible depending on context:

• després de classe — sometimes heard, but després de la classe is very standard
• quan s’acabi la classe = when class finishes
• en acabar la classe = on finishing class / after finishing class

For a learner, després de la classe is an excellent standard pattern to remember.