Learn Hardcore German: Am Wochenende entspannen wir uns zusammen im Park und hören Musik. - On the weekend we relax together in the park and listen to music.

German main clauses follow the rule: the conjugated verb is always in second position (the “V2 rule”).
“Second position” means second element, not second word.

You can put something other than the subject in first position, for example a time phrase:

Am Wochenende (1st element: time)
entspannen (2nd element: conjugated verb)
wir (3rd element: subject)

So:

Wir entspannen uns am Wochenende… (subject in first position)
and
Am Wochenende entspannen wir uns… (time in first position)

are both correct. They just emphasize different things. The second one emphasizes when it happens.

What exactly does Am Wochenende mean, and why is it am instead of an dem?

Am Wochenende means “on the weekend / at the weekend.”

Grammatically:

an = on/at
dem = the (dative, masculine/neuter singular)
Wochenende = weekend (neuter noun, das Wochenende)

an + dem contracts to am in spoken and written German:

an dem Wochenende → am Wochenende

This is the standard, natural way to say “on the weekend” in German.
Am Wochenende is a time expression in the dative case.

Why do we say entspannen wir uns and not just entspannen wir?

entspannen can be used:

Reflexively: sich entspannen = “to relax (oneself), to relax”
Transitively: etw. entspannen = “to relax something” (e.g. “to relax a muscle, to ease a situation”)

When talking about people relaxing, everyday German almost always uses the reflexive form:

Wir entspannen uns. = We relax (ourselves) / We relax.

The uns is the reflexive pronoun (for wir).
Other forms:

ich entspanne mich
du entspannst dich
er/sie/es entspannt sich
wir entspannen uns
ihr entspannt euch
sie/Sie entspannen sich

You can say wir entspannen without uns, but then it tends to sound more like “we relax something” or more formal/technical. In normal speech about people relaxing, wir entspannen uns is the idiomatic form.

Where does the reflexive pronoun uns usually go in the sentence? Could I move it?

With one conjugated verb in a main clause, the reflexive pronoun normally comes right after the verb:

Am Wochenende entspannen wir uns zusammen im Park.

Basic order (ignoring the initial time phrase):

entspannen (verb)
wir (subject)
uns (reflexive pronoun)
then other stuff (adverbs, place, etc.)

You could say:

Am Wochenende entspannen wir uns im Park zusammen.

This is also correct, but wir uns should stay very close to the verb. The pronoun normally doesn’t move far away from its verb.

Why is it im Park and not in den Park?

in can take either dative or accusative, depending on meaning:

dative → location (where something is)
accusative → direction (where something goes)

Here, “we relax in the park” describes a location (we are already in the park), so we use dative:

in + dem Park (dative) → im Park

If you talk about movement into the park, you’d use accusative:

Wir gehen in den Park. = We go to/into the park.

So:

im Park = in the park (location, dative)
in den Park = into the park (direction, accusative)

What is im exactly?

im is a contraction of:

in + dem → im

dem is the dative singular of der for masculine and neuter nouns.

der Park (nominative)
dem Park (dative) → im Park

Similar common contractions:

in + das → ins (ins Kino)
an + dem → am (am Wochenende)
bei + dem → beim
zu + dem → zum
zu + der → zur

Why is it just hören Musik and not hören die Musik or hören Musik an?

No article (“Musik” without die)

In German, when you talk about activities in a general, habitual way, especially with things like:

Musik hören (listen to music)
Fußball spielen (play football)
Klavier spielen (play piano)

you often omit the article:

Wir hören Musik. = We listen to music. (general)
Using die Musik would refer more to some specific music that was already mentioned.

No preposition “to”

German often uses the bare verb + object where English uses “to”:

Musik hören = to listen to music
There is no preposition like English “to” here.

Why not “anhören”?

(sich) etw. anhören is more like “to listen to something attentively / to give something a listen” and is often reflexive:

Wir hören uns ein Lied an. = We listen to a song.

But for the general activity “we listen to music”, Musik hören is the normal, neutral expression.

Why is it und hören Musik and not und wir hören Musik?

When you connect two main-clause verb phrases with und, German usually:

Mentions the subject once, then
Omits it in the second part if it’s the same subject.

So:

Am Wochenende entspannen wir uns zusammen im Park und hören Musik.

Here, wir is clearly the subject for both verbs: entspannen and hören. Saying und wir hören Musik is not wrong, but it’s more natural and concise to leave out the second wir.

Both are grammatically correct:

Am Wochenende entspannen wir uns zusammen im Park und hören Musik.
Am Wochenende entspannen wir uns zusammen im Park, und wir hören Musik. (slightly more emphasis on the second part)

Can this sentence also refer to the future (“This weekend we will relax…”) even though it’s in the present tense?

Yes. German often uses the present tense for near future events when the time is clear from context or from a time expression like am Wochenende.

So:

Am Wochenende entspannen wir uns zusammen im Park und hören Musik.

can mean:

“We relax together in the park and listen to music on weekends” (habit), or
“This coming weekend we’ll relax together in the park and listen to music” (plan).

If you want to stress the future, you can use Futur I:

Am Wochenende werden wir uns zusammen im Park entspannen und Musik hören.

But in everyday speech, the plain present with a clear time phrase is very common for the near future.

Where does zusammen usually go, and can I move it around?

zusammen is an adverb meaning together. In this sentence it is placed before the place phrase:

… entspannen wir uns zusammen im Park …

General tendencies:

It often comes close to the verb or before the place:
- Wir arbeiten zusammen.
- Wir wohnen zusammen in Berlin.
- Wir entspannen uns zusammen im Park.

You can move it:

Wir entspannen uns im Park zusammen. (also correct, a bit different rhythm)
Wir zusammen entspannen uns im Park. (unusual; native speakers normally wouldn’t say it like this)

The version in your sentence, zusammen im Park, is a very natural order.

If I want to say “We like to relax together in the park and listen to music”, where would I put gern?

gern (or gerne) usually stands in the “middle field”, after the verb and pronouns, before most adverbs/objects.

In your sentence:

Am Wochenende entspannen wir uns gern zusammen im Park und hören gern Musik.

Possible placements:

Am Wochenende entspannen wir uns gern zusammen im Park und hören Musik.
(we like the relaxing; listening to music is just stated)
Am Wochenende entspannen wir uns zusammen im Park und hören gern Musik.
(we like the music-listening in particular)
Am Wochenende entspannen wir uns gern zusammen im Park und hören gern Musik.
(we like both activities)

Basic rule: gern goes near the verb whose action you “like”, typically after the subject/pronoun and before most other information.