Learn Hardcore German: Ich laufe heute langsam im Park.

So with ich, the correct form is laufe.
Lauf by itself is usually an imperative (a command): Lauf! = Run! / Walk!
Läuft is the 3rd person singular form (er läuft = he runs).

Does laufen here mean to walk or to run? How do I know?

Laufen can mean to run or to walk, depending on context, region, and sometimes speaker preference.

In many parts of Germany, especially the north, laufen usually suggests running / jogging.
In some southern regions (e.g. parts of Switzerland, southern Germany), laufen can more casually mean to walk.

In a neutral textbook context, Ich laufe heute langsam im Park is often understood as I am jogging / running slowly in the park today.
If you clearly want to say walk, gehen is safest:

Ich gehe heute langsam im Park spazieren. – I’m going for a slow walk in the park today.

Why is the order heute langsam im Park and not something like im Park langsam heute?

German has a common guideline for adverbial order: Time – Manner – Place (often called the TMP rule).

In your sentence:

heute = time (when?)
langsam = manner (how?)
im Park = place (where?)

So Ich laufe heute langsam im Park follows Time – Manner – Place.

Other orders are possible, but they tend to sound less neutral and more marked, or they put emphasis on a particular element. For example:

Ich laufe im Park heute langsam. – possible, but sounds a bit unusual; emphasizes im Park first.
Ich laufe langsam heute im Park. – also possible, but not the neutral, default order.

For clear, natural German, heute langsam im Park is the best choice here.

Could I also say Heute laufe ich langsam im Park? Does that change the meaning?

Yes, that is perfectly correct: Heute laufe ich langsam im Park.

Differences:

Ich laufe heute langsam im Park. – neutral, subject ich comes first.
Heute laufe ich langsam im Park. – still neutral, but heute is moved to the first position, so it’s mildly emphasized: Today (as opposed to some other day) I’m walking / running slowly in the park.

The core meaning doesn’t change; it’s more about what you highlight.
In German main clauses, the finite verb must stay in the second position, so if Heute (or anything else) goes first, the verb (laufe) must come next, and ich moves after it:

Heute (1st position)
laufe (2nd position – the verb)
ich (3rd position – the subject)

Why is it im Park and not in dem Park?

Im is simply the contracted form of in dem.

in
- dem (dative masculine/neuter) → im

Park is masculine in German: der Park.
After in with a stationary meaning (in = inside, located in), masculine nouns take the dative case:

in dem Park → contracted to im Park

Both in dem Park and im Park are grammatically correct, but in everyday speech and writing, im is much more common and sounds more natural here.

Why is Park capitalized?

In German, all nouns are capitalized, no matter where they appear in the sentence.

Park is a noun (a thing, a place), so it must be written Park, not park.

This rule is strict and one of the most visible differences from English spelling.

Why is it im Park (dative) instead of in den Park (accusative)?

The choice between im Park and in den Park depends on whether you describe:

Location (where?) → usually dative
Direction / movement to a place (where to?) → usually accusative

In your sentence:

Ich laufe heute langsam im Park.
→ You are already in the park and moving within it. That’s a location: wo? (where?) → dative → im Park.

Compare:

Ich laufe heute langsam in den Park.
→ You are moving into the park (from outside to inside). That’s direction: wohin? (where to?) → accusative → in den Park.

Is langsam an adjective or an adverb here? How is it used?

In Ich laufe heute langsam im Park, langsam functions as an adverb: it describes how you are running/walking.

German often uses the same form for adjectives and adverbs:

ein langsamer Hund – a slow dog (adjective, agrees with the noun)
Der Hund läuft langsam. – The dog runs slowly. (adverb, describes the verb)

So:

As an adjective, it changes its ending to match the noun’s case, gender, and number.
As an adverb, it stays in its base form (langsam) and modifies the verb.

Where would I put nicht if I want to negate this sentence?

It depends on what exactly you want to negate. Some common options:

General negation of the activity in the park
- Ich laufe heute nicht im Park.
  → Today I’m not running/walking in the park. (Maybe I’m running somewhere else or not at all.)
Negating the slowness:
- Ich laufe heute nicht langsam im Park.
  → Today I’m not running slowly in the park (maybe fast).
Negating the whole predicate in a looser, more general way:
- Ich laufe heute nicht.
  → I’m not running/walking today.

Position of nicht is fairly flexible but usually comes before the element it negates, or near the end of the clause if it negates the entire predicate.

Can I say Ich bin heute langsam im Park am Laufen to express “I am walking slowly in the park”?

You can say Ich bin heute im Park am Laufen, and people will understand it, but:

The am-Progressiv (am Laufen sein) exists in spoken German (especially in some regions), but it’s less standard than English continuous forms.
In most contexts, plain present tense (Ich laufe heute langsam im Park) already covers the English “I am walking/running…”.

If you want to closely mirror the English right now feeling, a more standard way is to add gerade:

Ich laufe gerade langsam im Park. – I am (right now) walking/running slowly in the park.

Is it possible to move langsam to another position, like Ich laufe langsam heute im Park?

Yes, langsam can move, but the sentence may sound less neutral or change the emphasis.

Some possibilities:

Ich laufe heute langsam im Park. – neutral, follows Time–Manner–Place.
Ich laufe langsam heute im Park. – grammatically okay, but unusual; sounds like you’re emphasizing langsam or speaking with a non‑standard rhythm.
Ich laufe heute im Park langsam. – also possible, with a bit more emphasis on langsam, as if contrasting slow vs. fast.

The safest, most natural choice for normal speech or writing is Ich laufe heute langsam im Park.

Could I use another verb like gehen or spazieren here? What would change?

Yes, you can use other verbs with slightly different nuances:

gehen (to go / to walk):
- Ich gehe heute langsam im Park.
  → Grammatically correct, but sounds a bit incomplete; usually you’d say what kind of going:
- Ich gehe heute langsam im Park spazieren.
  → I’m going for a slow walk in the park today. (very natural)
spazieren (to stroll):
- Ich spaziere heute langsam im Park.
  → I’m taking a slow stroll in the park today. (calm, leisurely walking)
laufen:
- Neutral verbs for moving on foot, often understood as running/jogging in many regions.

So you choose the verb according to whether you want to stress walking, strolling, or running/jogging.

How do I pronounce laufe correctly?

Key points for laufe:

au: pronounced like English “ow” in “cow” → [laʊ]
f: like English f
e at the end: a short, unstressed uh sound, like the -e in “the” in very fast speech → [ə]

Approximate pronunciation: [ˈlaʊfə] → “LOW-fuh” (with LOW as in “cow”, not as in “low price”).

Combined:
Ich laufe heute langsam im Park.
→ Focus on: lau- as “low” (rhyming with how), not as English low.