Learn Hardcore Finnish: Opin hengittämään rauhallisemmin harjoittelemalla meditaatiota joka aamu. - I learn to breathe more calmly by practising meditation every morning.

In this sentence with harjoittelemalla meditaatiota joka aamu (by practising meditation every morning), it is most naturally understood as past:
I learned to breathe more calmly by practising meditation every morning.

Why is it hengittämään and not the basic infinitive hengittää?

Hengittämään is the third infinitive illative form of hengittää.

Verb: hengittää = to breathe
Stem for the 3rd infinitive: hengittä-
Illative ending: -mään (because of vowel harmony: ä → mään, not maan)
Result: hengittämään

The structure oppia + -maan/-mään form means to learn to do something (as an activity/process):

Opin hengittämään rauhallisemmin
= I learned to breathe more calmly (learned how/into the skill of breathing calmly).

You can also say:

Opin hengittää rauhallisemmin.

With oppia, both hengittää and hengittämään are possible. Using hengittämään often sounds a bit more like you are acquiring the ability/skill, not just the idea.

How is rauhallisemmin formed, and why not rauhallisesti?

Base adjective: rauhallinen = calm.

From this you get:

Comparative adjective: rauhallisempi = calmer
Comparative adverb: rauhallisemmin = more calmly

Pattern:

hiljainen (quiet) → hiljaisempi → hiljaisemmin (more quietly)
nopea (fast) → nopeampi → nopeammin (faster / more quickly)
rauhallinen (calm) → rauhallisempi → rauhallisemmin (more calmly)

Rauhallisesti would just mean calmly (no comparison).
Rauhallisemmin means more calmly – compared to before, or compared to some other way of breathing.

So:

Opin hengittämään rauhallisesti
= I learned to breathe calmly. (just calm)
Opin hengittämään rauhallisemmin
= I learned to breathe more calmly. (compared to before)

What is harjoittelemalla, and why does it end in -malla?

Harjoittelemalla is the third infinitive adessive of harjoitella (to practise).

Formation:

Verb: harjoitella
Stem for the 3rd infinitive: harjoittele-
Adessive ending: -malla (because of vowel harmony: a → malla, not mällä)
Result: harjoittelemalla

The -malla / -mällä form typically means “by doing X” (method or means):

Opin suomea puhumalla = I learn Finnish by speaking.
Paransin kuntoani juoksemalla = I improved my fitness by running.
Opin hengittämään rauhallisemmin harjoittelemalla meditaatiota
= I learned to breathe more calmly by practising meditation.

So harjoittelemalla answers “how? / by what means?”

Why is it meditaatiota and not meditaation?

Meditaatiota is the partitive singular of meditaatio (meditation).

The verb harjoitella usually takes its object in the partitive:

harjoitella pianoa = to practise (the) piano
harjoitella suomea = to practise Finnish
harjoitella meditaatiota = to practise meditation

Reasons for partitive here:

Harjoitella describes an ongoing, repeated activity, not a single completed act.
Partitive often marks incomplete / unbounded actions and “some amount of” something.

So meditaatiota suggests practising meditation as an ongoing skill, not “completing” one meditation like a finished object.

Meditaation (genitive) could occur in other structures (e.g. meditaation avulla = with the help of meditation), but not naturally as the direct object of harjoitella.

What exactly does joka aamu mean, and why is joka used with the singular?

Joka aamu literally is “which morning”, but as an idiom it means every morning.

With repeated time expressions, Finnish often uses joka + singular:

joka päivä = every day
joka viikko = every week
joka ilta = every evening
joka aamu = every morning

So even though in English we pluralise (every mornings), Finnish keeps the noun singular after joka.

You could also say jokainen aamu (each and every morning), which is more emphatic or slightly more formal.
In everyday speech, joka aamu is the most natural choice.

Can the word order be changed, for example putting joka aamu earlier in the sentence?

Yes, Finnish word order is fairly flexible. You can move adverbials like joka aamu around without changing the core meaning.

Original:

Opin hengittämään rauhallisemmin harjoittelemalla meditaatiota joka aamu.

Possible alternatives:

Joka aamu opin hengittämään rauhallisemmin harjoittelemalla meditaatiota.
Opin joka aamu hengittämään rauhallisemmin harjoittelemalla meditaatiota.
Opin hengittämään rauhallisemmin joka aamu harjoittelemalla meditaatiota.

They are all grammatical. The choice affects emphasis and rhythm, not the basic meaning.
The original version is natural and neutral in tone.

Is the subject minä missing here, and is that normal?

The subject minä is implied in the verb form opin.

In Finnish, personal pronouns are often dropped, because the verb ending already shows the person:

Opin hengittämään… = I learned to breathe…
Opit hengittämään… = You learned to breathe…
Opin and opit are enough; you don’t need to say minä or sinä.

You can say:

Minä opin hengittämään rauhallisemmin…

This adds emphasis on I (as opposed to someone else), but is not necessary in a neutral sentence.

Could this sentence also use the perfect tense, like Olen oppinut? What would be the difference?

Yes, you could say:

Olen oppinut hengittämään rauhallisemmin harjoittelemalla meditaatiota joka aamu.

Difference:

Opin… (simple past)
→ Focus on the learning event in the past.
At some point (over some period), I learned…
Olen oppinut… (present perfect)
→ Focus on the result that is relevant now.
I have (now) learned… and this skill still matters in the present.

Both are correct; you choose based on whether you want to tell a past story (opin) or highlight a present result (olen oppinut).

Can I replace harjoittelemalla with harjoittelemisen kautta or harjoittelemisen avulla, and how would that sound?

You can, but the nuance changes slightly.

harjoittelemalla meditaatiota
= by practising meditation (simple, direct way of expressing the means)

More explicit alternatives:

harjoittelemisen kautta = through practising
Opin hengittämään rauhallisemmin meditaation harjoittelemisen kautta.
harjoittelemisen avulla = with the help of practising
Opin hengittämään rauhallisemmin meditaation harjoittelemisen avulla.

These are grammatical but sound:

more formal or abstract, and
heavier than the very natural and compact harjoittelemalla meditaatiota.

For everyday language, harjoittelemalla is the best choice.