Learn Hardcore Finnish: Kurssi auttaa minua oppimaan suomea joka päivä. - The course helps me to learn Finnish every day.

minä = nominative (used for subjects: Minä opiskelen. – I study.)
minut = accusative/genitive-type form (used e.g. after näet: Näet minut. – You see me.)
minua = partitive (used here because auttaa selects partitive for the person helped)

Using minä or minut in this sentence would be ungrammatical in standard Finnish:
✗ Kurssi auttaa minä / minut oppimaan suomea joka päivä.

Why is the verb oppimaan and not oppia?

Finnish commonly uses the structure:

auttaa + (object in partitive) + verb in the -maan / -mään form

This -maan / -mään form is the illative of the third infinitive, and it often means “in doing / to do”:

auttaa minua oppimaan – help me (in) learning / to learn
auttaa sinua siivoamaan – help you (in) cleaning / to clean

So after auttaa, the natural pattern is:

auttaa + ketä? (whom?) + tekemään mitä? (doing what?)

Using the basic infinitive oppia here sounds wrong in standard Finnish:

✗ Kurssi auttaa minua oppia suomea joka päivä.

What exactly is the form oppimaan, and how is it formed?

Oppimaan is:

the illative case of the third infinitive of oppia (“to learn”).

Formation (simplified):

Take the strong stem of the verb: oppia → oppi-
Add -ma / -mä: oppi- + ma → oppima-
Add the illative ending -an / -än: oppima- + an → oppimaan

So:

syödä (to eat) → syömään
kirjoittaa (to write) → kirjoittamaan
puhua (to speak) → puhumaan

Typical meanings:

motion or direction: Menen nukkumaan. – I’m going to sleep.
purpose / activity: Aloitan opiskelemaan suomea. – I start studying Finnish.
with auttaa: Kurssi auttaa minua oppimaan. – The course helps me learn.

Why is it suomea and not suomi? What case is suomea?

Suomea is the partitive singular of suomi (“Finnish” as a language).

The verb oppia (“to learn”) usually takes its object in the partitive when the learning is seen as an ongoing / incomplete process or an unbounded quantity:

Haluan oppia suomea. – I want to learn Finnish. (in general)
Opin suomen vuodessa. – I learned Finnish in a year. (completed achievement)

So in your sentence:

oppimaan suomea = to learn Finnish (some Finnish, the language in general)

The bare suomi (Kurssi auttaa minua oppimaan suomi) would be wrong in this structure; you’d either use partitive suomea (normal) or, in special contexts where you mean “the whole language, fully mastered”, a total object like suomen:

Lopulta opin suomen. – I eventually learned Finnish (fully).

Could I say Kurssi auttaa minua oppia suomea joka päivä like in English “helps me to learn”?

No, that is not natural or correct Finnish.

With auttaa, the normal pattern is:

auttaa + (object) + verb in -maan / -mään, not the plain infinitive:

Correct:

Kurssi auttaa minua oppimaan suomea joka päivä.

Incorrect:

✗ Kurssi auttaa minua oppia suomea joka päivä.

Think of it more like “helps me with learning Finnish” or “helps me in learning Finnish” – that “with/in learning” idea is expressed by oppimaan, not oppia, in Finnish.

Why is joka päivä used, and can I move it in the sentence?

Joka päivä literally means “every day”:

joka – “every / each”
päivä – “day” (singular nominative)

Even though the meaning is plural (“every day”), Finnish uses singular here.

Word order is flexible. All of these are grammatically fine, with slightly different emphasis:

Kurssi auttaa minua oppimaan suomea joka päivä.
(neutral: focus on learning Finnish every day)
Kurssi auttaa minua joka päivä oppimaan suomea.
(bit more emphasis that the helping happens every day)
Joka päivä kurssi auttaa minua oppimaan suomea.
(strong emphasis on “every day”)

You can also replace joka päivä with e.g.:

päivittäin – daily
joka ikinen päivä – every single day (very emphatic)

What exactly does joka mean here, and why isn’t päivä in the plural?

Here joka is a determiner meaning “every / each”.

In this use:

joka + singular noun = “every X”

Examples:

joka päivä – every day
joka viikko – every week
joka kuukausi – every month

Finnish does not say joka päivää or joka päivät here. The singular form päivä is used even though English uses a plural noun (“every day”).

Can I leave out minua and just say Kurssi auttaa oppimaan suomea joka päivä?

Yes, you can.

Kurssi auttaa minua oppimaan suomea joka päivä.
– The course helps me learn Finnish every day. (specifically you)
Kurssi auttaa oppimaan suomea joka päivä.
– The course helps (people) learn Finnish every day. (general statement, “one”, “you”, “students”)

If the context already makes it clear who is being helped (e.g. you’re talking about a course you are taking), leaving out minua is quite natural.

Does kurssi mean “a course” or “the course”? Why is there no article?

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

The noun kurssi is just “course” in bare form. Whether you translate it as “a course” or “the course” depends entirely on context:

Kurssi auttaa minua oppimaan suomea joka päivä.
– A course helps me learn Finnish every day. (if you’re introducing it)
– The course helps me learn Finnish every day. (if it’s already known which course)

So the Finnish sentence itself does not specify definiteness; English has to choose “a” or “the” based on the situation.

What is the difference between oppia and opiskella in sentences like this?

Both relate to learning/studying, but they focus on different aspects:

oppia = to learn, to acquire knowledge/skill, focus on the result
- Haluan oppia suomea. – I want to learn Finnish.
opiskella = to study, focus on the activity/process
- Haluan opiskella suomea. – I want to study Finnish.

So you could also say:

Kurssi auttaa minua opiskelemaan suomea joka päivä.
– The course helps me (to) study Finnish every day.

Your original sentence with oppimaan emphasizes that the course helps you actually learn / make progress, not just do the activity of studying.

Why is suomea not capitalized, even though “Finnish” is capitalized in English?

In Finnish, names of languages are not capitalized unless they start a sentence.

Minä puhun suomea. – I speak Finnish.
Hän opiskelee ruotsia ja saksaa. – She studies Swedish and German.

The word Suomi (capital S) is the name of the country (Finland).
But the language in cases like suomea, suomi, suomeksi is written with a lowercase s.