Dopo il lavoro, ho bisogno di un po’ di tranquillità.

Because bisogno is a noun, not an adjective.
Italian expresses this idea as avere bisogno di..., literally to have need of....

So:

• ho bisogno di tranquillità = I need some peace/quiet
• literally: I have need of peace/quiet

That is why Italian uses ho (from avere, to have), not sono (from essere, to be).

In Italian, bisogno normally takes di before the thing that is needed.

Pattern:

• avere bisogno di + noun
• avere bisogno di + infinitive

Examples:

• Ho bisogno di acqua.
• Ho bisogno di dormire.

So in your sentence:

• ho bisogno di un po’ di tranquillità

the first di belongs to the expression avere bisogno di.

They come from two different structures:

1. ho bisogno di... = I need...
2. un po’ di... = a little / some ...

So:

• ho bisogno di = I need
• un po’ di tranquillità = a bit of peace/quiet

Put together:

• ho bisogno di un po’ di tranquillità

This is completely normal in Italian.

Un po’ di means a little, a bit of, or some, depending on context.

Examples:

• un po’ di tempo = a little time / some time
• un po’ di acqua = a little water
• un po’ di tranquillità = a little peace/quiet

It is a very common expression in everyday Italian.

Because po’ is a shortened form of poco.

So:

• poco → shortened to po’

The apostrophe shows that letters have been dropped.
This is why the correct spelling is po’, not pò.

A common learner mistake is writing un pò with an accent. Standard Italian uses un po’ with an apostrophe.

Because the stress falls on the final syllable: tran-quil-li-tà.

In Italian, words stressed on the last syllable usually take a written accent, so:

• città
• università
• tranquillità

The accent helps show both pronunciation and correct spelling.

Italian uses the definite article much more often than English, especially in expressions of time or routine.

So:

• dopo il lavoro = after work

Even though English often says just after work, Italian normally says dopo il lavoro.

This is very common:

• prima della scuola = before school
• dopo la cena = after dinner
• durante la lezione = during class

Usually, standard Italian prefers dopo il lavoro.

A bare noun like dopo lavoro can appear in some contexts, but it is less neutral and may sound more like a fixed expression or label than an ordinary sentence.

For a learner, the safest and most natural choice is:

• Dopo il lavoro...

Because Italian often drops subject pronouns when they are not needed.

The verb ending already tells you who the subject is:

• ho = I have
• hai = you have
• ha = he/she has

So ho bisogno already means I need / I have need.

You can say io ho bisogno if you want emphasis, for example:

• Io ho bisogno di tranquillità, non di rumore.
  I need peace and quiet, not noise.

But in a normal sentence, ho bisogno is enough.

Italian word order is flexible. Putting Dopo il lavoro first sets the time frame right away:

• Dopo il lavoro, ho bisogno di un po’ di tranquillità.

This is very natural and common.

You could also say:

• Ho bisogno di un po’ di tranquillità dopo il lavoro.

Both are correct. The first version emphasizes when this is true.

It is not strictly necessary, but it is very natural.

When a time phrase like Dopo il lavoro comes at the beginning, a comma often helps separate it from the main clause:

• Dopo il lavoro, ho bisogno di un po’ di tranquillità.

Without the comma, the sentence is still understandable:

• Dopo il lavoro ho bisogno di un po’ di tranquillità.

The comma mainly improves readability.

Because here Italian needs a noun, not an adjective.

• tranquillo / tranquilla = calm, quiet (adjective)
• tranquillità = calmness, peace, quiet (noun)

Since un po’ di must be followed by a noun, Italian uses:

• un po’ di tranquillità

not:

• un po’ di tranquillo

A simple approximate pronunciation is:

DOH-poh eel lah-VOH-roh, oh bee-ZOHN-yoh dee oom poh' dee trahn-kweel-lee-TAH

A few useful points:

• gli is not in this sentence, so nothing tricky there.
• gn in bisogno sounds like the ny in canyon.
• ll in tranquillità is pronounced clearly.
• the final stress in tranquillità falls on -tà.

If you want to sound natural, keep the rhythm smooth and let tranquillità end strongly on the last syllable.

Here dopo is a preposition, meaning after.

It introduces the noun phrase:

• dopo il lavoro = after work

You can also see dopo used as an adverb in other contexts, but in this sentence it is functioning as a preposition.

Yes. Both are grammatical, but the tone changes a little.

• Ho bisogno di tranquillità. = I need peace/quiet.
• Ho bisogno di un po’ di tranquillità. = I need a bit of peace/quiet.

Adding un po’ di makes the request sound a little softer, more natural, and often more conversational.