Learn Hardcore Greek: Προσπαθώ να μιλάω ήρεμα στη συνάντηση.

In Προσπαθώ να μιλάω, it links προσπαθώ (I try) with μιλάω (I speak), so the whole thing means “I try to speak”.
Grammatically, να + verb is called the subjunctive in Greek, especially after verbs like θέλω (I want), μπορώ (I can), προσπαθώ (I try), etc.

Why is it να μιλάω and not να μιλήσω?

Greek distinguishes between imperfective (ongoing / repeated) and perfective (single / completed) aspect.

να μιλάω focuses on how you speak during the meeting, as an ongoing manner or habit:
Προσπαθώ να μιλάω ήρεμα στη συνάντηση. = I try to (generally) speak calmly in the meeting.
να μιλήσω would sound more like a single act of speaking:
Προσπαθώ να μιλήσω ήρεμα. = I’m trying to speak calmly (this time, in this one instance).

So να μιλάω fits better if you mean your overall way of speaking in the meeting.

What’s the difference between μιλάω and μιλώ?

They are just two forms of the same verb, meaning I speak / I talk.

μιλάω is the more common, everyday spoken form.
μιλώ is slightly more formal or literary, but also correct.

You could say Προσπαθώ να μιλώ ήρεμα στη συνάντηση, and it would still be correct Greek, just a bit more formal in tone.

Why is ήρεμα used instead of ήρεμος?

Ήρεμα is an adverb, describing how you speak: calmly.
Ήρεμος is an adjective, describing a noun: a calm person / a calm voice, etc.

Μιλάω ήρεμα. = I speak calmly. (manner of speaking)
Έχω ήρεμη φωνή. = I have a calm voice. (property of the voice)

Because we are modifying the verb μιλάω, Greek uses the adverb form ήρεμα.

Can I say ήσυχα instead of ήρεμα? Do they mean the same?

Ήρεμα and ήσυχα are close, but not identical:

ήρεμα = calmly, without emotional agitation (opposite of nervous, upset, angry).
ήσυχα = quietly, without noise (opposite of loud).

So:

Μιλάω ήρεμα. = I speak in a calm, non‑aggressive way.
Μιλάω ήσυχα. = I speak quietly, not loudly.

In your sentence, ήρεμα is better if you mean emotionally calm, which fits “at the meeting” context well.

What exactly is στη in στη συνάντηση?

Στη is a contraction of two words:

σε (in, at, to)
τη (the – feminine, singular, accusative)

So:

σε τη συνάντηση → στη συνάντηση

This contraction is very regular in Greek:

σε + τη → στη(ν)
σε + τον → στον
σε + το → στο

Why is it τη συνάντηση? What gender is συνάντηση?

Συνάντηση is a feminine noun. Its main forms are:

η συνάντηση = the meeting (nominative)
τη(ν) συνάντηση = the meeting (accusative; used after σε)

Many nouns ending in -ση are feminine in Greek (e.g. η άσκηση, η κίνηση, η συζήτηση), so η συνάντηση follows that common pattern.

Can I omit the article and just say σε συνάντηση?

Yes, but the meaning changes slightly:

στη συνάντηση = at the meeting (a specific, known meeting).
σε συνάντηση = at a meeting / in a meeting (some meeting, not specified).

So if you mean a particular meeting (e.g. the Monday team meeting), στη συνάντηση is the natural choice.

Can I move the word ήρεμα or στη συνάντηση? For example: Προσπαθώ να μιλάω στη συνάντηση ήρεμα?

Yes. Greek word order is quite flexible.

All of these are grammatically fine:

Προσπαθώ να μιλάω ήρεμα στη συνάντηση.
Προσπαθώ να μιλάω στη συνάντηση ήρεμα.

The usual choice is to keep ήρεμα close to μιλάω, but moving it to the end is also acceptable and doesn’t really change the meaning here; it just slightly shifts the rhythm/emphasis.

How would the meaning change if I said Προσπάθησα να μιλήσω ήρεμα στη συνάντηση instead?

Προσπάθησα is past tense (I tried), and να μιλήσω is perfective (a single act of speaking).

Προσπαθώ να μιλάω ήρεμα στη συνάντηση.
= I try (in general / these days) to speak calmly in the meeting.
Προσπάθησα να μιλήσω ήρεμα στη συνάντηση.
= I tried to speak calmly in the meeting (on that occasion, in the past).

So you switch from a present, ongoing habit/effort to a specific attempt in the past.