Després de treballar, buscaré la meva amiga al barri.

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

• després de + noun → after + noun
• després de + infinitive → after doing something

So:

• després de treballar = after working

Without de, the phrase would sound incomplete or ungrammatical here.

After després de, Catalan uses the infinitive to express an action in a general way, just like English often uses after working, after eating, after studying.

So:

• treballar = to work / working in this structure

You are not saying a fully conjugated clause such as after I work; you are using the verbal idea working.

Buscaré is the simple future tense, first person singular:

• buscar = to look for
• buscaré = I will look for

Catalan often forms the future by adding endings directly to the infinitive:

• buscaré = I will look for
• buscaràs = you will look for
• buscarà = he/she will look for
• buscarem = we will look for
• buscareu = you all will look for
• buscaran = they will look for

The accent in buscaré also shows the stress falls on the last syllable.

Because Catalan buscar takes a direct object.

In English, you say:

• look for my friend

But in Catalan, the structure is more like:

• buscar la meva amiga = look for my friend

So Catalan does not need an extra preposition here. This is very important for English speakers, because it is easy to copy the English structure by mistake.

In Catalan, possessives are very often used together with the definite article:

• el meu amic
• la meva amiga
• els meus llibres

So la meva amiga is the normal pattern for my friend.

English does not use an article before my, but Catalan usually does.
There are some exceptions, especially with certain family terms in some contexts, but with amiga, la meva amiga is the natural form.

Because amiga is feminine singular.

The possessive has to agree with the noun:

• el meu amic = my male friend
• la meva amiga = my female friend

So:

• meu = masculine singular
• meva = feminine singular

Since amiga is feminine, the sentence uses meva.

Al is the contraction of a + el.

• a
  • el = al

Since barri is masculine singular (el barri), Catalan contracts them:

• al barri

This is very common:

• al cotxe = to/in the car
• al carrer = to/in the street
• al barri = to/in the neighborhood

Here it means the place where the action happens, so it is understood as in the neighborhood or around the neighborhood.

Even though a often corresponds to English to, Catalan also uses a / al in many location expressions where English would use in or at.

So in this sentence, al barri tells you where the speaker will look for the friend.

The comma is natural because després de treballar is an introductory time phrase at the beginning of the sentence.

So the sentence is divided like this:

• Després de treballar, = time frame
• buscaré la meva amiga al barri. = main clause

In short sentences, punctuation can sometimes vary a little, but the comma here is standard and helpful.

A simple pronunciation guide:

• Després → stress on the second syllable: des-PRÉS
• treballar → stress on the last syllable: tre-ba-LLAR
• buscaré → stress on the last syllable: bus-ca-RÉ
• barri → stress on the first syllable: BAR-ri

A few useful notes:

• The ll in treballar is the Catalan ll sound.
• The accent in després and buscaré marks the stressed syllable.
• rr in barri is a strong rolled or trilled r in careful pronunciation.

Yes, Catalan word order is somewhat flexible, but this version is very natural and neutral:

• Després de treballar, buscaré la meva amiga al barri.

You could move the time phrase, for example:

• Buscaré la meva amiga al barri després de treballar.

That still works, but the original sentence puts the time information first, which is very common.