Trebam novu pastu za zube, pa idem u drogeriju.

Trebam is the 1st person singular present form of trebati.

So:

• trebati = to need
• trebam = I need

In this sentence, trebam novu pastu za zube means I need a new toothpaste.

Croatian does not have articles like English a/an/the.

So novu pastu za zube can mean:

• a new toothpaste
• the new toothpaste

The exact meaning depends on context. English has to choose an article; Croatian usually does not.

Because the noun is the direct object of trebam, so it has to be in the accusative case.

The basic dictionary form is:

• nova pasta = nominative, new toothpaste

But after trebam:

• trebam novu pastu = accusative, I need new toothpaste

Both the adjective and the noun change:

• nova → novu
• pasta → pastu

The preposition za here means for, and in this use it takes the accusative.

So:

• zub = tooth
• zube = teeth, accusative plural

That is why Croatian says:

• pasta za zube = literally paste for teeth

This is the normal Croatian way to say toothpaste.

Yes. Both are correct and natural.

• pasta za zube = toothpaste
• zubna pasta = tooth-paste / dental paste

Pasta za zube is very common in everyday speech.
Zubna pasta is also common and may sound a little more compact or label-like.

Here pa means something like:

• so
• and so
• then

It connects the two ideas:

• I need new toothpaste
• so I’m going to the drugstore

It is a very common Croatian connector in everyday speech.

Croatian often uses the present tense for something that is happening now or is an immediate, intended next action.

So idem u drogeriju can mean:

• I’m going to the drugstore
• I’m heading to the drugstore
• I go to the drugstore depending on context

In this sentence, it sounds natural because the second action follows directly from the first.

If you wanted to make the future more explicit, you could say something like:

• Trebam novu pastu za zube, pa ću ići u drogeriju.

Because Croatian uses different cases after u depending on whether you mean movement or location.

• u + accusative = motion into / to
• u + locative = being in

So:

• Idem u drogeriju = I’m going to the drugstore
• U drogeriji sam = I am in the drugstore

That is a very important Croatian pattern.

Not exactly. This is an important vocabulary point.

Drogerija is more like:

• drugstore
• a shop for toiletries, cosmetics, shampoo, toothpaste, cleaning products, and similar items

A true pharmacy where you get medicine is usually:

• ljekarna in Croatian

So if you need toothpaste, drogerija makes perfect sense.

Yes, absolutely. That is also very natural.

Compare:

• Trebam novu pastu za zube.
• Treba mi nova pasta za zube.

Both mean roughly I need a new toothpaste.

Very roughly:

• trebam + object is a direct I need X
• treba mi + noun is more like X is needed by me, but in real usage it is just a normal way to say I need X

Both are common.

Croatian often omits subject pronouns because the verb ending already shows who the subject is.

For example:

• trebam already tells you it means I need
• idem already tells you it means I go / I’m going

So ja is usually unnecessary unless you want emphasis or contrast:

• Ja trebam novu pastu, a ne ti. = I need new toothpaste, not you.

Because the sentence contains two clauses, each with its own verb:

• Trebam novu pastu za zube
• pa idem u drogeriju

In Croatian, when pa connects two full clauses like this, a comma is normally used.

So the comma helps show the structure:

• first idea: I need new toothpaste
• second idea: so I’m going to the drugstore