Learn Hardcore Croatian: Cijeli tjedan učimo hrvatski navečer. - All week we study Croatian in the evening.

Since učimo already means we (are) learn(ing), saying mi učimo is usually redundant.
You would add mi only for emphasis, e.g. Mi učimo hrvatski, a oni ne. – We study Croatian, but they don’t.

What tense is učimo here? Does it mean “we study” or “we are studying” or “we have been studying”?

Učimo is present tense of učiti (imperfective aspect). Croatian present can cover several English meanings, depending on context:

We study Croatian (regular/habitual)
We are studying Croatian (right now / this period)
We have been studying Croatian (all week) (ongoing activity over a period up to now)

Because of cijeli tjedan (the whole week), the sentence is naturally understood as something happening throughout this week, similar to English We’ve been studying Croatian in the evenings all week or We’re studying Croatian in the evenings all week.

What case is cijeli tjedan, and why is it used?

Cijeli tjedan is in the accusative singular:

nominative: cijeli tjedan
accusative: cijeli tjedan (same form for masculine inanimate)

This is an example of the accusative of duration: Croatian uses the accusative without a preposition to express how long something lasts.

Examples:

Učimo cijeli tjedan. – We study the whole week.
Radio je cijeli dan. – He worked (for) the whole day.
Čekali smo sat vremena. – We waited for an hour.

So cijeli tjedan answers “for how long?”

Why is hrvatski not capitalized? In English, “Croatian” is.

In Croatian, names of languages are not capitalized. They are written with a small letter:

hrvatski – Croatian
engleski – English
njemački – German
španjolski – Spanish

Proper nouns like country and nationality names do take a capital letter:

Hrvatska – Croatia (country)
Hrvat, Hrvatica – a Croat (male / female)

So učimo hrvatski is the standard spelling.

Is hrvatski an adjective here (“Croatian”), or a noun (“Croatian language”)? Why is there no word jezik?

Formally, hrvatski is an adjective meaning Croatian, but in this context it is used as a noun, meaning Croatian (language). That’s very common with language names in Croatian.

Full forms would be:

Učimo hrvatski jezik. – We study the Croatian language.
Učimo hrvatski. – We study Croatian. (the word jezik is understood)

Both are correct; Učimo hrvatski is more natural in everyday speech.

Can I change the word order, like Učimo hrvatski cijeli tjedan navečer or Navečer učimo hrvatski cijeli tjedan?

Croatian word order is fairly flexible, and all of these can be grammatically correct, but the emphasis changes.

Some common variants:

Cijeli tjedan učimo hrvatski navečer.
Neutral; slight emphasis on the whole week.
Učimo hrvatski cijeli tjedan navečer.
More neutral order: verb + object, then time expressions. Sounds fine.
Navečer učimo hrvatski cijeli tjedan.
Emphasizes in the evening (contrast with some other time of day).
Hrvatski učimo cijeli tjedan navečer.
Emphasizes Croatian (as opposed to some other subject).

In practice, most natural are 1 and 2; others are used when you want to stress a particular element.

Why is navečer one word? Could I say na večer or u večer instead?

Navečer is an adverb meaning in the evening / at night (evenings), and it’s normally written as one word.

navečer – in the evening
uvečer – also “in the evening”; similar meaning, a bit less common regionally

Phrases like na večer or u večer are not the normal way to express “in the evening” as a general time expression. You might see na večer in certain fixed or poetic expressions, but for everyday speech you should use:

navečer or uvečer – in the evening
ujutro – in the morning
popodne / poslijepodne – in the afternoon
noću – at night

Why is there no preposition for “in the evening”? In English, we say “in the evening”, but Croatian just has navečer.

Croatian often uses adverbs of time instead of prepositional phrases:

navečer / uvečer – in the evening
ujutro – in the morning
popodne / poslijepodne – in the afternoon
noću – at night

These behave like single words (adverbs), so no extra preposition is needed.
So instead of something like u večer, Croatian normally uses navečer or uvečer.

What is the difference between cijeli tjedan and svaki tjedan?

cijeli tjedan = (for) the whole week / all week (this week, that specific week)
→ Focus on duration within one specific week.
svaki tjedan = every week / each week
→ Repeated action over many weeks.

Examples:

Cijeli tjedan učimo hrvatski navečer.
We’re studying Croatian in the evenings all this week.
Svaki tjedan učimo hrvatski navečer.
We study Croatian in the evenings every week (regular habit).

Is there a difference between cijeli tjedan and čitav tjedan?

In this context, cijeli tjedan and čitav tjedan are practically synonyms: both mean the whole week / all week.

Cijeli tjedan učimo hrvatski navečer.
Čitav tjedan učimo hrvatski navečer.

Stylistically:

cijeli is slightly more neutral and common.
čitav can sound a bit more expressive or emphatic in some contexts, but the difference is small.

Can Cijeli tjedan učimo hrvatski navečer also mean a future plan: “All next week we will be studying Croatian in the evening”?

Yes, Croatian present tense can refer to the near future when there is a clear time expression:

Sutra učimo hrvatski. – Tomorrow we’re (going to be) studying Croatian.
U petak pišemo test. – On Friday we’re writing a test.

So Cijeli tjedan učimo hrvatski navečer can mean:

a description of what is happening this week, or
a scheduled plan for an upcoming week.

Context (for example, whether you’re talking about this week or next week) will clarify the time.