Después de cenar, me gusta salir a tomar el aire un rato.

Both are possible, but they mean slightly different things in structure:

• después de cenar = after having dinner / after dinner
  Here, cenar is an infinitive verb.
• después de la cena = after dinner
  Here, la cena is a noun.

In Spanish, después de + infinitive is very common when English would often use after + -ing:

• después de comer = after eating / after lunch
• después de trabajar = after working

So después de cenar is a very natural way to say it.

Because after después de, Spanish often uses an infinitive when the action is general and the subject is understood.

So:

• después de cenar literally works like after dining / after eating dinner

English usually uses after + -ing or just after dinner, but Spanish prefers the infinitive here.

More examples:

• después de estudiar = after studying
• después de ducharse = after showering
• después de salir = after going out / after leaving

Because gustar works differently from English to like.

Spanish structures it more like:

• me gusta salir = going out is pleasing to me

So:

• me = to me
• gusta = is pleasing
• salir a tomar el aire un rato = going out to get some air for a while

That is why Spanish says me gusta instead of something that directly matches I like.

A few comparisons:

• Me gusta el café = I like coffee
• Me gustan los libros = I like books
• Me gusta salir = I like going out

Because the thing being liked here is treated as a single action: salir a tomar el aire un rato.

With gustar, the verb agrees with what comes after it grammatically, not with the person who likes it.

• Me gusta salir → salir is an infinitive, so Spanish uses gusta
• Me gustan los paseos → los paseos is plural, so Spanish uses gustan

So in this sentence, the whole idea of going out to get some air for a while is one singular thing, which is why gusta is correct.

It is a very natural expression meaning:

• to go out to get some air
• often understood as to go out for some fresh air

Breaking it down:

• salir = to go out / to leave
• a = to
• tomar el aire = literally to take the air

This is idiomatic Spanish. In natural English, you would not usually say take the air, but in Spanish it sounds normal.

Because salir a + infinitive means to go out in order to do something.

So:

• salir a pasear = to go out for a walk
• salir a correr = to go out to run
• salir a comprar pan = to go out to buy bread
• salir a tomar el aire = to go out to get some air

Here, a introduces the purpose of going out.

Yes, pretty much. It is an idiomatic expression.

Literally, it is to take the air, but in real usage it means:

• to get some air
• to get some fresh air
• sometimes simply to step outside for a bit

It is a common, natural phrase. Learners should understand it as a chunk rather than translating each word separately every time.

Un rato means for a while or for a bit.

So:

• salir a tomar el aire un rato = to go out to get some air for a while

It is a very common time expression in Spanish.

Examples:

• Voy a descansar un rato. = I’m going to rest for a while.
• Espera un rato. = Wait a bit.
• Hablamos un rato. = We talked for a while.

It often gives the sentence a casual, everyday feel.

Because in Spanish, short time expressions like un rato often come after the verb phrase they modify.

So the order:

• salir a tomar el aire un rato

sounds very natural.

You could think of it as:

• to go out to get some air for a bit

Spanish often places this kind of expression toward the end:

• Voy a leer un rato.
• Vamos a caminar un poco.
• Se quedó allí un momento.

Technically, salir can mean both to leave and to go out, depending on context. But in this sentence, go out is the natural meaning.

Because of a tomar el aire, the idea is clearly:

• stepping outside
• going out for some air

So leave to get some air is possible as a literal interpretation, but go out to get some air is the best natural translation.

Because Spanish often leaves out subject pronouns when they are already clear from the verb or context.

So instead of saying:

• Yo, después de cenar, me gusta salir...

Spanish normally just says:

• Después de cenar, me gusta salir...

The sentence already clearly means I like... because of me gusta.

In fact, adding yo here would usually sound unnecessary unless you were emphasizing contrast, for example:

• A mí me gusta salir, pero a mi hermano no.
  = I like going out, but my brother doesn’t.

The comma separates the introductory time phrase from the rest of the sentence:

• Después de cenar, ...

It works like English:

• After dinner, I like to go out...

In short sentences, Spanish punctuation can sometimes be flexible, but here the comma is very natural because después de cenar sets the scene before the main clause.