Learn Hardcore Finnish: Puhetyöpaja on minulle tärkeä. - The speech workshop is important to me.

So puhetyöpaja is literally a speech‑workshop, usually meaning something like a speaking workshop, public speaking workshop, or speech therapy workshop depending on context.

Why is it minulle and not minä or minun?

Minä is the basic subject form (I), and minun is the genitive (my).
In this sentence we need to/for me, not I or my, so Finnish uses a case ending instead of a separate word:

minä = I
minun = my
minulle = to me / for me

The idea is “The speech workshop is important to me”, so minulle is the correct form.

What case is minulle, and what does that case usually express?

Minulle is in the allative case.

The allative usually expresses movement onto, to, or towards something, and more abstractly the idea of “to/for someone”.
With adjectives like tärkeä (important), Finnish typically uses the allative to mark the person who feels that importance:

tärkeä minulle = important to/for me
tärkeä sinulle = important to/for you
tärkeä meille = important to/for us

So the allative case is how Finnish handles what English expresses with the preposition to or for here.

Why is tärkeä in the basic form? Shouldn’t it change to agree with minulle?

In this sentence, tärkeä agrees with the subject, not with minulle.

Subject: Puhetyöpaja (singular)
Predicative adjective describing it: tärkeä (singular nominative)

Minulle is just an extra phrase telling us to whom it is important. Finnish predicate adjectives agree in number and (when needed) case with the subject, not with the person affected:

Puhetyöpaja on tärkeä. – The workshop is important.
Puhetyöpajat ovat tärkeitä. – The workshops are important.

So tärkeä stays in its basic form because puhetyöpaja is one thing in the basic (nominative) form.

Can I change the word order to Puhetyöpaja on tärkeä minulle or Minulle puhetyöpaja on tärkeä?

Yes, both are possible and correct; Finnish word order is fairly flexible.

Puhetyöpaja on minulle tärkeä. (neutral, common)
Puhetyöpaja on tärkeä minulle. (also neutral; just places the emphasis slightly more on tärkeä)
Minulle puhetyöpaja on tärkeä. (emphasizes to me, as in “for me, the workshop is important”)

All three are understandable and natural; the differences are mostly about subtle emphasis and rhythm, not about grammar.

How would I say “The speech workshops are important to me”?

You need a plural subject and a plural form of the verb and adjective:

Puhetyöpajat ovat minulle tärkeitä.

Breakdown:

Puhetyöpajat = the speech workshops (plural nominative)
ovat = are (3rd person plural of olla, “to be”)
minulle = to me / for me
tärkeitä = important (plural partitive form, very common in such plural predicatives)

So Puhetyöpajat ovat minulle tärkeitä = The speech workshops are important to me.

What is the function of on here? Is it just like English “is”?

Yes. On is the 3rd person singular present form of the verb olla (to be).

olla = to be
hän / se on = he/she/it is
Puhetyöpaja on = The speech workshop is

In this sentence, on is a linking verb (copula) connecting the subject (Puhetyöpaja) with the adjective (tärkeä). It works very much like English “is”.

Could I say Puhetyöpaja on minulle hyvin tärkeä or erittäin tärkeä?

Yes. You can easily add adverbs to intensify tärkeä:

Puhetyöpaja on minulle hyvin tärkeä. – The speech workshop is very important to me.
Puhetyöpaja on minulle erittäin tärkeä. – The speech workshop is extremely / very important to me.

The structure stays the same; you just place the adverb before tärkeä.

How do I pronounce puhetyöpaja, and where is the stress?

In Finnish, the primary stress is always on the first syllable of a word:

PU‑he‑TYÖ‑pa‑ja

Pronunciation tips:

pu = “poo” but shorter
he = like “heh”
työ = front-rounded vowel; the yö is like trying to say “eu” in French “deux”, but as one syllable
pa = “pa” as in “papa”
ja = “ya” (Finnish j sounds like English y in yes)

Every written vowel and consonant is pronounced; there are no silent letters.

Is there a difference between minulle tärkeä and minulle tärkeä asia?

Yes, but the difference is mostly whether you name the “thing” explicitly.

minulle tärkeä = important to me (with the noun understood from context)
minulle tärkeä asia = a thing that is important to me / an issue that is important to me

In the sentence Puhetyöpaja on minulle tärkeä, the “thing” is clearly puhetyöpaja, so adding asia would be redundant. You’d normally just say tärkeä there, not tärkeä asia.