Learn Hardcore Croatian: Plastika je najveći problem za naš planet. - Plastic is the biggest problem for our planet.

Plastika can mean plastic, the plastic, or plastics depending on context.
naš planet can mean our planet or the/our planet, again decided only by context.

Definiteness is usually clear from meaning, context, or word order, not from a separate word.

What does je mean here, and can I leave it out?

je is the 3rd person singular present of biti (to be). It links the subject to what is being said about it:

Plastika (subject)
je (is)
najveći problem za naš planet (predicate)

In standard Croatian full sentences you normally must include je.
You can drop je only in very telegraphic styles, like headlines or slogans, e.g. on a poster, but in normal speech and writing you say Plastika je najveći problem…, not Plastika najveći problem…

Why is problem in the form problem, not something like problema or problemom?

problem is in the nominative singular, because after je (the verb to be) Croatian usually uses the nominative for nouns that describe the subject (this is called a predicate noun):

Plastika – nominative (subject)
je – is
najveći problem – nominative (what plastic is)

You do not say Plastika je najveći problemom in modern standard Croatian.
Genitive (problema) or instrumental (problemom) are used in other structures, but not in this simple X is Y sentence.

Why is it najveći problem, not najveća problem to agree with plastika (which is feminine)?

The adjective najveći agrees with problem, not with plastika.

The structure is:

Subject: Plastika (feminine)
Verb: je
Predicate noun phrase: najveći problem (masculine)

Inside the predicate phrase, najveći describes problem, which is a masculine noun, so the adjective must be masculine nominative singular:

masculine: najveći problem
feminine: najveća planeta
neuter: najveće pitanje

So Plastika je najveći problem is correct; Plastika je najveća problem is wrong.

How is the word najveći formed, and what are the basic forms related to it?

najveći is the superlative of velik (big, large):

positive: velik – big
comparative: veći – bigger
superlative: najveći – biggest

The pattern is:

take the comparative (veći)
add the prefix naj- → najveći

Other examples:

brz → brži → najbrži (fast → faster → fastest)
pametan → pametniji → najpametniji (smart → smarter → smartest)

Why do we use za in za naš planet, and what case does it take?

za here means for in the sense of to the detriment/benefit of:

problem za naš planet = a problem for our planet

In this meaning, za is followed by the accusative case.
So you get:

za naš planet (for our planet) – planet is masculine accusative singular
za našu planetu (for our planet) – planetu is feminine accusative singular
za naš grad (for our city)
za našu djecu (for our children)

Why is it naš planet and not našeg planeta? Aren’t we talking about our planet?

There are actually two different structures you can use:

With za + accusative (as in your sentence):
- najveći problem za naš planet
  = the biggest problem for our planet
Here za already expresses for, so the noun is in the accusative: naš planet.
With a genitive (no za):
- najveći problem našeg planeta
  = the biggest problem of our planet
Here našeg planeta is genitive (of our planet).

Both are correct but have slightly different structures and prepositions. In your sentence, because you have za, you use the accusative: za naš planet, not za našeg planeta.

Why does naš look the same in naš planet as in the dictionary form? Shouldn’t it change for case?

naš does change for case, gender, and number, but in this particular case the form happens to coincide with the basic dictionary form.

planet is masculine inanimate.
For masculine inanimate nouns, the accusative singular = nominative singular.

So:

nominative: naš planet (our planet – as subject)
accusative: naš planet (for our planet – after za)

Compare with an animate noun:

nominative: naš pas (our dog)
accusative: za našeg psa (for our dog)

So naš does inflect, but here accusative and nominative simply look the same.

What is the difference between planet and planeta? Could I say za našu planetu?

Both planet and planeta are used in Croatian and mean planet.

planet – masculine noun
planeta – feminine noun

So both are correct:

za naš planet (masc accusative: naš planet)
za našu planetu (fem accusative: našu planetu)

They sound very similar in meaning here; style preferences and region may influence which one is more common, but grammatically both options are fine.

Why is plastika singular if we are talking about plastic in general, not one specific piece?

In Croatian, material nouns like plastika, voda (water), zlato (gold) are usually treated as mass nouns and appear in the singular when you talk about the substance in general:

Plastika je skupa. – Plastic is expensive.
Voda je važna. – Water is important.

You can use the plural plastike when you mean types or kinds of plastic:

Razne plastike se koriste u industriji. – Various plastics are used in industry.

In your sentence, we mean plastic as a material in general, so the singular plastika is correct.

Can I change the word order, for example to Najveći problem za naš planet je plastika?

Yes. Both word orders are correct:

Plastika je najveći problem za naš planet.
Najveći problem za naš planet je plastika.

The difference is mainly in emphasis:

Starting with Plastika emphasizes plastic as the topic.
Starting with Najveći problem… emphasizes the problem and then identifies it as plastic.

This kind of flexible word order is common in Croatian; grammar is shown mostly by endings, not by fixed word order.

How do you pronounce je and ć in najveći?

Pronunciation tips:

j is pronounced like English y in yes.
- je sounds like ye in yes.
ć is a soft sound, somewhat like the t
- y in British when said quickly, or a softer ch.
- najveći is roughly nai-ve-chi, but the ch is softer and more palatal than in English.

So the whole sentence:

Plastika je najveći problem za naš planet.
Approximate: PLAH-stee-kah yeh NAI-veh-chee PRO-blehm zah nash PLAH-net.