Learn Hardcore Greek: Προχωράμε αργά στο πάρκο και χαλαρώνουμε.

The ending -με in προχωράμε and χαλαρώνουμε tells you the subject is “we”.
Εμείς (we) is only added for emphasis or contrast, e.g.
- Εμείς προχωράμε αργά, αυτοί τρέχουν.
  We walk slowly, they run.

So the sentence doesn’t need an explicit “we”; it’s understood from the verb.

What’s the difference between προχωράμε, πηγαίνουμε, and περπατάμε?

All can involve movement, but they have different nuances:

προχωράμε
Literally “we go forward / we advance”.
Neutral “moving on” or “walking on”, often implying progress, continuing along a path.
πηγαίνουμε (from πηγαίνω)
“We go” (in general, by any means: walking, car, bus, etc.). Focus on going to a place.
περπατάμε (from περπατάω/περπατώ)
“We walk” (specifically on foot).

In your sentence:

Προχωράμε αργά στο πάρκο και χαλαρώνουμε.
Suggests “We’re moving/walking slowly in the park and relaxing,” with a feeling of proceeding calmly.

If you said:

Περπατάμε αργά στο πάρκο και χαλαρώνουμε.
That would more explicitly mean “We walk slowly in the park and relax.”

What does αργά mean exactly, and what kind of word is it?

αργά means “slowly” and is an adverb.

It comes from the adjective αργός, -ή, -ό (slow).
The adverb is formed from the neuter plural form:

αργός → αργά (slowly)
καλός → καλά (well)
γρήγορος → γρήγορα (quickly)

In the sentence:

Προχωράμε αργά = We move slowly / We walk slowly.

Why is it στο πάρκο and not something like just πάρκο? What does στο mean?

στο is a contraction of the preposition and article:

σε (in, at, to) + το (the, neuter singular) → στο

So στο πάρκο literally is “in/to the park”.

Greek uses the definite article (ο, η, το) much more than English, even with general places:

στο πάρκο = in the park / to the park
στο σχολείο = at school / to school
στο σπίτι = at home / to the house

You could say σε ένα πάρκο = in a park, but the default everyday way is with the article: στο πάρκο.

Does στο πάρκο here mean “in the park” or “to the park”?

Literally, σε + accusative can mean in / at / to, and context decides.

In this sentence:

Προχωράμε αργά στο πάρκο και χαλαρώνουμε.

Because it’s followed by “and relax”, the most natural understanding is:

“We walk/move slowly in the park and relax.”

If you wanted to highlight direction “to the park,” you might add words or context:

Προχωράμε αργά προς το πάρκο.
We move/walk slowly toward the park.

Can I change the word order? For example, is Στο πάρκο προχωράμε αργά και χαλαρώνουμε also correct?

Yes, Greek word order is fairly flexible, especially for adverbs and prepositional phrases. All of these are grammatically correct, with small differences in emphasis:

Προχωράμε αργά στο πάρκο και χαλαρώνουμε.
Neutral: We walk slowly in the park and relax.
Στο πάρκο προχωράμε αργά και χαλαρώνουμε.
Slight emphasis on “In the park” – maybe contrasting with some other place.
Προχωράμε στο πάρκο αργά και χαλαρώνουμε.
Emphasis a bit more on how (slowly) after saying where.

The basic meaning stays the same; Greek uses word order partly for focus/emphasis, not just for grammar.

In English we distinguish “we walk” vs “we are walking.” Does Greek have this difference in the present tense?

In the present tense, Modern Greek does not grammatically distinguish:

simple present (we walk, we relax)
vs
present continuous (we are walking, we are relaxing)

Both are expressed with the same form:

Προχωράμε = we walk or we are walking
Χαλαρώνουμε = we relax or we are relaxing

Context (or adverbs like τώρα = now, συχνά = often) makes the meaning clear, not a separate verb form.

Is χαλαρώνουμε a reflexive verb? Why don’t we say something like “relax ourselves”?

No extra reflexive pronoun is needed here. Χαλαρώνω is simply:

χαλαρώνω = I relax (become relaxed, unwind)

In Greek, many verbs that would be reflexive in other languages are just intransitive:

Ξυπνάω = I wake up (no “myself”)
κουράζομαι = I get tired (this one is middle/reflexive in form)
χαλαρώνω = I relax

So χαλαρώνουμε already means we relax / we are relaxing without adding anything like ourselves.

Why is there a definite article with πάρκο? Could I say just σε πάρκο?

Greek tends to use the definite article much more than English, especially with common locations and concrete nouns.

στο πάρκο = in the park (often meaning “in the park” in general, not a specific one you’ve just mentioned)
στο σχολείο = at school
στη δουλειά = at work

You can say σε ένα πάρκο = in a park, which sounds more like “in some (unspecified) park.”
But bare σε πάρκο (without an article or ένα) is usually not natural here. You almost always want:

σε ένα πάρκο (a park)
or
στο πάρκο (the park).

How do you pronounce Προχωράμε αργά στο πάρκο και χαλαρώνουμε? Where is the stress?

Stress marks in Greek show which syllable is emphasized:

Προχωράμε → pro-cho-RA-me (stress on ρά)
αργά → ar-GA
στο → sto (one syllable)
πάρκο → PAR-ko (stress on παρ)
και → ke (pronounced like “keh”)
χαλαρώνουμε → cha-la-RO-nou-me (stress on ρών)

A rough phonetic guide (using English approximations):

πρ like “pr”
ο like o in lot
χ like German “ch” in Bach (a hard, breathy “h” at the back of the throat)
ου like oo in food
α like “a” in father
κ like “k”
ρ is a rolled/trilled “r”

I’ve seen both προχωράμε and προχωρούμε. Which one is correct?

Both are accepted forms of the 1st person plural present of προχωράω/προχωρώ:

προχωράμε
προχωρούμε

In everyday Modern Greek:

-άμε forms (προχωράμε, μιλάμε, δουλεύουμε) are more colloquial and common in speech.
-ούμε / -ούμε forms are slightly more formal or written, but still understood and used.

In your sentence, Προχωράμε αργά στο πάρκο και χαλαρώνουμε is perfectly natural modern spoken Greek.