Μετά από μια δύσκολη μέρα, προσπαθώ να είμαι ήρεμος, αναπνέοντας βαθιά.

Μετά από means after followed by a time expression or event: μετά από μια δύσκολη μέρα = after a difficult day.

• μετά από + noun is the normal modern Greek way to say after [something].
• μετά on its own usually means afterwards / then (an adverb, not a preposition):
  
  • Μετά πήγα σπίτι. = Afterwards I went home.

You might see μετά directly before a noun in more formal or older styles, but in everyday modern Greek you should use μετά από before a noun phrase.

• μέρα is feminine (η μέρα, της μέρας, etc.), so:
  • article: μια (indefinite feminine)
  • adjective: δύσκολη (feminine form of δύσκολος)

So μια δύσκολη μέρα literally = a difficult day.

About the variants:

• μια vs μία:

  • Both are pronounced the same in modern Greek.
  • μια is the usual everyday spelling for a / one (fem.).
  • μία is often used when you want to stress one (as a number) or in more formal writing.

• μέρα vs ημέρα:

  • μέρα is the normal everyday word for day.
  • ημέρα is more formal / written (official language, set phrases).

The comma separates the introductory time phrase from the main clause:

• Μετά από μια δύσκολη μέρα, = After a difficult day, (setting the time)
• προσπαθώ να είμαι ήρεμος... = I try to be calm...

In Greek, it is normal to put a comma after an initial phrase like Μετά από μια δύσκολη μέρα (time, place, condition, etc.) before the main sentence that follows. It’s similar to English punctuation in this case.

Greek is a pro‑drop language: the subject pronoun is usually omitted because it is clear from the verb ending.

• προσπαθώ already means I try (first person singular).
• Adding εγώ (Εγώ προσπαθώ) is possible, but it adds emphasis, like I (myself) try or I, on my part, try.

So the natural, neutral version is just προσπαθώ without εγώ.

In modern Greek, many verbs (like προσπαθώ, θέλω, μπορώ, etc.) are followed by a να‑clause, not by a bare verb.

• Structure: προσπαθώ + να + verb
  
  • προσπαθώ να είμαι ήρεμος = I try to be calm

Να marks a kind of subjunctive / non‑finite clause (things that are wished, tried, planned, possible, etc.).
Without να, προσπαθώ είμαι is ungrammatical in modern Greek.

Both can be translated as I try to stay/remain calm, but the aspect is a bit different:

• προσπαθώ να είμαι ήρεμος

  • Focus on a more ongoing state: I try to be / to keep being calm in general or over a period of time.

• προσπαθώ να μείνω ήρεμος (from μένω)

  • Focus more on remaining / staying calm at a specific moment or through a particular situation.

In many everyday contexts they can overlap and both sound natural; να είμαι ήρεμος is slightly more general and state‑oriented, να μείνω ήρεμος slightly more event‑focused.

Ήρεμος is an adjective and must agree with the (understood) subject I in gender and number.

Here, the default is a male speaker or a generic masculine form:

• masculine singular: ήρεμος
• feminine singular: ήρεμη
• neuter singular: ήρεμο
• masculine plural: ήρεμοι
• feminine plural: ήρεμες
• neuter plural: ήρεμα

So:

• A man: προσπαθώ να είμαι ήρεμος
• A woman: προσπαθώ να είμαι ήρεμη
• Several people (mixed / all male): προσπαθούμε να είμαστε ήρεμοι
• Several women: προσπαθούμε να είμαστε ήρεμες

Αναπνέοντας is the present active participle of the verb αναπνέω (to breathe).

• Formed roughly as: αναπνέ‑
  • ‑οντας → αναπνέοντας
• It usually corresponds to an English ‑ing form used adverbially:
  • προσπαθώ να είμαι ήρεμος, αναπνέοντας βαθιά
    = I try to be calm, breathing deeply / by breathing deeply.

Some key points:

• It shares the subject with the main verb (I try, I breathe).
• In modern Greek it is mostly used as an adverbial participle (describing how/when/why), not as a noun.
  
  • For Breathing is important, you would say Η αναπνοή είναι σημαντική, not Το αναπνέοντας είναι σημαντικό.

The comma marks αναπνέοντας βαθιά as an additional, descriptive phrase:

• ..., προσπαθώ να είμαι ήρεμος, αναπνέοντας βαθιά.
  → I try to be calm, breathing deeply.

With the comma, the participle phrase is slightly more detached, like extra information about how you try to be calm.

In many cases, you could omit the comma:

• προσπαθώ να είμαι ήρεμος αναπνέοντας βαθιά

Then it reads as more tightly integrated: I try to be calm by breathing deeply.
Both versions are acceptable; the comma simply highlights the phrase more as a side comment / manner description.

Βαθιά here means deeply, so αναπνέοντας βαθιά = breathing deeply.

Grammatically:

• βαθύς = deep (adjective, masculine)
• βαθιά can be:
  • the feminine form of the adjective (μια βαθιά λίμνη = a deep lake), or
  • an adverb meaning deeply / deep (αναπνέω βαθιά = I breathe deeply).

In αναπνέοντας βαθιά, it is an adverb modifying αναπνέοντας (how you are breathing).

Both adjectives can relate to calmness, but they are not always interchangeable:

• ήρεμος: calm, serene, peaceful (emotionally or outwardly)

  • προσπαθώ να είμαι ήρεμος = I try to stay calm (not stressed, not upset).

• ήσυχος: quiet, not noisy, also undisturbed / not bothered

  • μια ήσυχη γειτονιά = a quiet neighborhood
  • Άφησέ με ήσυχο. = Leave me alone.

You can say προσπαθώ να είμαι ήσυχος, but it tends to sound more like I try to be quiet / not cause trouble rather than specifically emotionally calm. For the meaning of the given sentence, ήρεμος is the better choice.