Learn Hardcore Croatian: Večeras planiramo miran izlazak u park. - Tonight we are planning a calm outing to the park.

All are correct. The usual, neutral position is at the start or just after the verb, but moving it can add slight emphasis (e.g. putting it at the end can stress when more).

Why is "planiramo" in the present tense if the action is in the future (this evening)?

In Croatian, the present tense is often used to talk about very near future plans, especially when there is a time expression like večeras, sutra (tomorrow), etc.

Večeras planiramo miran izlazak u park.
= Tonight we’re planning a quiet outing to the park / Tonight we plan…

Using future tense (ćemo planirati, we will plan) would sound strange here, because the planning is already decided now. If you wanted to focus on the going rather than the planning, you’d more likely say:

Večeras ćemo ići u park. – We will go to the park tonight.

Why is it "miran izlazak" and not "mirno izlazak" or something else?

Miran is an adjective meaning "peaceful / quiet / calm", and it has to agree in gender, number, and case with the noun it describes.

izlazak – masculine, singular, accusative (as a direct object)
Adjective must match: miran izlazak (masc. sg. acc.)

For a masculine inanimate noun, the accusative form is the same as the nominative, so:

nominative: miran izlazak
accusative: miran izlazak

Mirno is mostly the neuter form or an adverb ("quietly"), so mirno izlazak is incorrect here.

Why is it "u park" and not "u parku"?

The preposition u can take:

Accusative – when there is motion into something ("to/into")
Locative – when you are already in a place ("in/at")

Here we have u park with the accusative because we’re going to the park:

miran izlazak u park – a quiet outing to the park (motion towards)

Compare:

Večeras smo u parku. – Tonight we are in the park. (locative: parku)
Večeras idemo u park. – Tonight we’re going to the park. (accusative: park)

In English we say “to the park”. Why doesn’t Croatian use something like a word for “the” in "u park"?

Croatian has no articles (no "a", "an", "the"). The noun form park with u simply covers both English possibilities:

u park can mean to a park or to the park
u parku can mean in a park or in the park

Definiteness is usually understood from context, word order, or is just not as important to mark as in English.

Can I change the word order, for example: "Planiramo miran izlazak u park večeras"? Is that still correct?

Yes, that word order is grammatically correct:

Planiramo miran izlazak u park večeras.

Croatian word order is relatively flexible. All of these are okay, with slight changes in emphasis:

Večeras planiramo miran izlazak u park. (neutral, common)
Planiramo večeras miran izlazak u park. (emphasis a bit more on večeras)
Planiramo miran izlazak u park večeras. (slight focus on večeras at the end)

The key is to keep words that belong together close (e.g. miran izlazak, u park).

What exactly does "izlazak" mean here? Is it just “going out”?

Izlazak is a noun from the verb izaći / izlaziti (to go out). It means:

"going out", "outing", "going out socially"

In everyday speech, izlazak often implies going out for social reasons:

izlazak u grad – going out to town (often: bars, cafes, etc.)
izlazak u park – going out to the park

So miran izlazak u park = a quiet, low-key outing to the park.

If you used a verb instead, you might say:

Večeras idemo u park. – We’re going to the park tonight.

Where is the word for “we” in "planiramo"? How do I know it means “we plan”?

In Croatian, personal pronouns are usually dropped because the verb ending already shows the person:

planiram – I plan
planiraš – you (sg.) plan
planira – he/she/it plans
planiramo – we plan
planirate – you (pl.) plan
planiraju – they plan

So planiramo by itself means “we plan / we are planning”.
You could optionally add the pronoun for emphasis:

Mi večeras planiramo miran izlazak u park. – stressing we, not someone else.

How do the adjective and noun endings in "miran izlazak" relate to gender and case?

izlazak – masculine, singular
In accusative (as the direct object) for masculine inanimate nouns, the form is the same as nominative: izlazak

The adjective miran agrees with izlazak:

masculine, singular, accusative → miran izlazak

If the noun changed, the adjective would change:

mirna večer (feminine singular) – a quiet evening
mirno mjesto (neuter singular) – a quiet place
mirne večeri (plural feminine) – quiet evenings

Here, both are masculine singular, so: miran izlazak.

Could I say "mirni izlazak" instead of "miran izlazak"?

In standard Croatian, with izlazak as a direct object in this kind of neutral sentence, miran izlazak is the normal, correct form.

Mirni izlazak is the masculine singular nominative (also used in some other contexts), which might appear in different structures, for example:

Mirni izlazak u park bio je odličan.
The quiet outing to the park was great.
(Here mirni izlazak is the subject, nominative case.)

So:

As a direct object (what we are planning): planiramo miran izlazak
As a subject: mirni izlazak je bio…