Learn Hardcore Finnish: Juoksu puistossa on hyvä tapa rentoutua. - Running in the park is a good way to relax.

juoksu = running / a run / the act of running (as a thing)
It is in the nominative singular and acts as the subject of the sentence.
Literally: Juoksu puistossa = Running in the park
on = is
hyvä tapa rentoutua = a good way to relax.

So grammatically the structure is:
[subject] Juoksu puistossa + [verb] on + [predicate] hyvä tapa rentoutua.

Why is it juoksu and not juokseminen?

Both are possible, but they differ slightly in nuance:

juoksu
- A “lexical” noun meaning running (as a sport/activity), a run.
- Often sounds a bit more sporty or activity-like, a hobby:
  - Juoksu on hyvä harrastus. – Running is a good hobby.
juokseminen
- The -minen form is a more “verbal” noun: the act of running, the activity of running.
- Slightly more neutral and very common as a generic action:
  - Juokseminen puistossa on hyvä tapa rentoutua.

In your sentence, Juoksu puistossa on hyvä tapa rentoutua is perfectly natural and maybe a bit more like “Running (as an activity) in the park is a good way to relax.”
Juokseminen puistossa on hyvä tapa rentoutua would also be correct and fully idiomatic.

Could I say Juosta puistossa on hyvä tapa rentoutua instead?

No, that sounds wrong in Finnish.

Finnish does not normally use the bare infinitive (juosta) as a subject the way English uses “to run”:

❌ Juosta puistossa on hyvä tapa rentoutua.
✅ Juoksu puistossa on hyvä tapa rentoutua.
✅ Juokseminen puistossa on hyvä tapa rentoutua.

As a subject, Finnish prefers either:

A noun like juoksu, or
A -minen form like juokseminen.

The plain infinitive juosta is used mostly:

after certain verbs (e.g. Haluan juosta – I want to run),
or in some fixed patterns, but not as a straightforward subject in sentences like this.

What case is puistossa, and what does it add to the meaning?

puistossa is in the inessive case (ending -ssa / -ssä), which usually means “in” or “inside/within” something.

puisto = park
puistossa = in the park / inside the park / at the park

So Juoksu puistossa literally means running in the park (as opposed to on a track, at home, etc.).
The inessive here functions as a location adverbial: it tells you where the running happens.

Why is there no article like “the” in puistossa? Should it mean “in the park” or “in a park”?

Finnish has no articles (no “a/an” or “the”).

The word puistossa by itself can be translated as either:

in the park
in a park

Which one you choose in English depends entirely on the context, not on Finnish grammar.
In this kind of generic sentence, English usually prefers:

Running in the park is a good way to relax.

but Finnish does not force a distinction here.

What is the function of on in this sentence?

on is the 3rd-person singular present tense form of the verb olla (to be).

olla = to be
on = is

It links the subject Juoksu puistossa to the predicate hyvä tapa rentoutua:

Juoksu puistossa (subject)
on (is)
hyvä tapa rentoutua (predicate / complement)

What does tapa mean here, and why is it used?

tapa is a noun meaning:

way, manner
habit, custom

In this sentence:

hyvä tapa rentoutua = a good way to relax

This is a very common and natural structure in Finnish:

hyvä tapa oppia – a good way to learn
huono tapa syödä – a bad way to eat
tapa puhua – a way of speaking

So tapa is used to express the idea of “method/means of doing something.”

What form is rentoutua, and why is it used after tapa?

rentoutua is the basic (dictionary) form of the verb, traditionally called the first infinitive.

After the noun tapa, Finnish typically uses this basic infinitive to express “a way to do X”:

tapa rentoutua – a way to relax
tapa oppia – a way to learn
tapa säästää rahaa – a way to save money

Finnish does not add a separate word for “to” (like English does). The infinitive ending -a / -ä combined with the structure already gives the meaning “to relax” in this context.

Can the word order be changed, for example: Puistossa juoksu on hyvä tapa rentoutua?

Yes, you can change the word order, but it affects the emphasis:

Juoksu puistossa on hyvä tapa rentoutua.
- Neutral emphasis on running in the park as a whole.
Puistossa juoksu on hyvä tapa rentoutua.
- Emphasis shifts to puistossa (in the park), as if contrasting it with other places:
  - Running in the park (as opposed to somewhere else) is a good way to relax.

Both are grammatically correct, but the original Juoksu puistossa on hyvä tapa rentoutua is the most natural neutral version.

Is Juoksu puistossa a complete phrase on its own, or does it always need a verb like on?

Juoksu puistossa is a noun phrase, not a complete sentence.
On its own, it just names an activity:

Juoksu puistossa – running in the park

To make a full sentence, you still need a verb:

Juoksu puistossa on kivaa. – Running in the park is fun.
Juoksu puistossa on hyvä tapa rentoutua. – Running in the park is a good way to relax.
Pidän juoksusta puistossa. – I like running in the park. (here the phrase is split: juoksusta
- puistossa)

Could I say Hyvä tapa rentoutua on juosta puistossa instead, and what’s the difference?

Yes, that is also correct and natural:

Hyvä tapa rentoutua on juosta puistossa.
= A good way to relax is to run in the park.

The main differences are:

Original: Juoksu puistossa on hyvä tapa rentoutua.
- Focus: Running in the park (what kind of habit it is).
Alternative: Hyvä tapa rentoutua on juosta puistossa.
- Focus: A good way to relax (and then we explain what it is: to run in the park).

So the meaning is almost the same, but the topic/emphasis changes slightly. The grammar also changes:

In the original, juoksu is a noun as subject.
In the alternative, juosta is an infinitive used in a complement clause after on.