Learn Hardcore Croatian: Glavni cilj ovog tečaja je da učiš hrvatski bez stresa. - The main goal of this course is for you to learn Croatian without stress.

So Glavni cilj ovog tečaja literally means “The main goal of this course”.
If you said Glavni cilj ovaj tečaj, it would be ungrammatical: you’d have two nominatives in a place where you need a genitive to express possession/association.

What case is “ovog tečaja”, exactly, and when is that case used?

Ovog tečaja is in the genitive singular:

ovog – genitive singular masculine of ovaj (“this”)
tečaja – genitive singular masculine of tečaj (“course”)

The genitive is used here mainly to show possession or belonging:

cilj ovog tečaja – the goal of this course
učenici ovog tečaja – the students of this course
naziv knjige – the title of the book
(knjige is also genitive)

So the pattern is similar to English “the X of Y”, but Croatian marks it with case endings instead of a separate word like of.

Why does the sentence use “je da učiš” and not just an infinitive like “je učiti”?

Croatian has two common ways to express goals or purposes after cilj je (“the goal is”):

Infinitive construction (more neutral/standard):
- Glavni cilj ovog tečaja je učiti hrvatski bez stresa.
- “The main goal of this course is to learn Croatian without stress.”
“da” + present tense construction (also very common, often a bit more personal or colloquial):
- Glavni cilj ovog tečaja je da učiš hrvatski bez stresa.
- Literally: “The main goal of this course is that you learn Croatian without stress.”

In your sentence, the teacher is directly addressing “you”, so “da učiš” sounds personal and motivating.
Using je učiti would be perfectly correct, just slightly less direct toward the listener.

What does “da” do in “da učiš”? Is it like English “that”?

Yes, here da is a conjunction similar to English “that” introducing a subordinate clause:

da učiš hrvatski bez stresa
≈ “that you learn Croatian without stress”

In Croatian, “da + present tense” often plays the role that “that + (subjunctive/infinitive)” plays in English:

Želim da naučiš hrvatski. – “I want you to learn Croatian.”
Cilj je da učiš bez stresa. – “The goal is (that) you learn without stress.”

So da marks a purpose/wish/intention clause rather than plain factual description.

Why is it “učiš” (2nd person singular) and not something like “uči” or “učiti”?

Učiš is second person singular present tense of učiti (“to learn”):

ja učim – I learn
ti učiš – you learn (singular, familiar)
on/ona/ono uči – he/she/it learns

The sentence is written as if the course is talking directly to you:

cilj … je da ti učiš hrvatski – “the goal … is that you learn Croatian”

Alternatives:

je učiti hrvatski – “is to learn Croatian” (impersonal, abstract)
je da se uči hrvatski – “is that Croatian is learned” / “for Croatian to be learned” (more impersonal)

So učiš is chosen to address the learner personally.

What’s the difference between “učiš” and “naučiš” here? Could we say “da naučiš hrvatski”?

Yes, you could say:

Glavni cilj ovog tečaja je da naučiš hrvatski bez stresa.

The difference is about aspect (ongoing process vs completed result):

učiti (imperf.) – to be learning / to study (focus on the process)
naučiti (perf.) – to learn, to master (focus on the result, having learned)

So:

da učiš hrvatski – the goal is that you are learning Croatian (you engage in the process, study regularly).
da naučiš hrvatski – the goal is that you end up knowing Croatian (reach a certain level).

Both make sense; naučiš emphasizes achieving mastery, učiš emphasizes learning without stress as an ongoing activity.

Why is it just “hrvatski” and not “hrvatski jezik” or “hrvatskog”?

Hrvatski here is the direct object of učiš, so it must be in the accusative case.

Verb: učiti – “to learn”
Question: Što učiš? – “What are you learning?”
Answer: (Učim) hrvatski. – “(I’m learning) Croatian.”

In Croatian, you can say:

učiti hrvatski – learn Croatian (short form)
učiti hrvatski jezik – learn the Croatian language (more explicit)

Hrvatski is an adjective (“Croatian”), but here it’s used as a noun (“Croatian [language]”), with jezik understood from context.

You would not use hrvatskog here, because hrvatskog is genitive (or accusative masculine animate); you need accusative inanimate, which for masculine is the same as the nominative: hrvatski.

Is “hrvatski” in this sentence an adjective or a noun?

Formally, hrvatski is an adjective (“Croatian”), masculine singular.

Functionally, here it behaves like a noun, because the noun jezik (“language”) is omitted but understood:

Full form: učiti hrvatski jezik
Usual shortened form: učiti hrvatski

Croatian very often uses adjectives like this as nouns when the noun is obvious:

govorim engleski (jezik) – I speak English (language)
učim njemački (jezik) – I’m learning German (language)

Why is it “bez stresa” and not something like “bez stres” or “bez stresu”?

Because the preposition bez (“without”) always takes the genitive case in Croatian.

bez + genitive

Here:

stres – nominative singular (dictionary form)
stresa – genitive singular

So:

bez stresa – without stress
bez problema – without problems (problem → problema, genitive)
bez šećera – without sugar (šećer → šećera, genitive)

“bez stres” or “bez stresu” would be ungrammatical.

Can the word order be changed, for example “Glavni cilj ovog tečaja je da bez stresa učiš hrvatski”?

Yes, Croatian word order is fairly flexible, and your variant is grammatical:

Glavni cilj ovog tečaja je da učiš hrvatski bez stresa.
(neutral, slight focus on learning Croatian)
Glavni cilj ovog tečaja je da bez stresa učiš hrvatski.
(slight focus on without stress)

You could also hear:

Glavni cilj ovog tečaja je da hrvatski učiš bez stresa.
Glavni cilj ovog tečaja je da učiš bez stresa hrvatski. (less natural, but possible in speech for emphasis)

The basic rule: keep all parts of the “da”-clause together, but you can reorder hrvatski and bez stresa to change emphasis. The original sentence is the most neutral and natural.

Why is “glavni” before “cilj” and “ovog” before “tečaja”? Is that the normal adjective order?

Yes. In Croatian, adjectives and demonstratives normally come before the nouns they modify:

glavni cilj – the main goal
ovaj tečaj – this course
ovog tečaja – of this course
dobar prijatelj – a good friend
taj problem – that problem

If there are multiple modifiers, the usual order is:

Demonstrative / possessive (ovaj, taj, moj, tvoj…)
Descriptive adjective (glavni, dobar, novi…)
Noun

For example:

ovaj novi tečaj – this new course
moj glavni cilj – my main goal

In your sentence, there is just one adjective per noun, so it’s straightforward:

glavni (adj.) + cilj (noun)
ovog (demonstrative adj.) + tečaja (noun)

Could we drop “je” and say “Glavni cilj ovog tečaja da učiš hrvatski bez stresa”?

In standard written Croatian, you should not drop je here. The copula (“to be”) is required:

Glavni cilj ovog tečaja je da učiš hrvatski bez stresa. ✔

In informal spoken language, people sometimes drop je in certain positions, so you might hear something like:

Glavni cilj ovog tečaja, da učiš hrvatski bez stresa.

But in correct, neutral writing, always include je in this type of sentence.