Learn Hardcore Croatian: Zvat ću ju sutra.

Use the short infinitive (drop final -i) when the auxiliary follows: zvat ću, pisat ću, radit ću.
Use the full infinitive when the auxiliary comes before: ću zvati, ću pisati, ću raditi. Writing zvati ću is incorrect; it must be either zvat ću or ću zvati.

Is ju the same as je? When do I use nju?

je and ju are both standard unstressed forms for the feminine singular accusative (her). You can say Zvat ću je or Zvat ću ju; both are correct.
ju is very common in everyday Croatian; je is also fine. Choice is mostly stylistic/euphonic.
Use the stressed form nju when the pronoun is emphasized or cannot be a clitic (e.g., at sentence start or after a preposition): Nju ću zvati, ne njega. / Za nju ću glasati.

Can I move words around? What are other natural orders?

Yes. Keep clitics (ću, je/ju) in their proper slots, but you can shift elements for emphasis:

Zvat ću je/ju sutra. (neutral)
Ja ću je/ju zvati sutra. (emphasizes “I”)
Sutra ću je/ju zvati. (emphasizes “tomorrow”)
Nju ću zvati sutra. (emphasizes “her”) Avoid ungrammatical orders like Zvat je ću or starting with Ju ću....

Where exactly does ću go? Why is it after the verb here?

ću is a clitic that prefers second position in the clause. In Zvat ću ju sutra, the first stressed word is Zvat, so ću follows it. If you place another word first, ću follows that: Sutra ću ju zvati. You can also put the auxiliary first and keep the full infinitive: ću zvati je/ju sutra (more commonly ću je/ju zvati).

Is there any difference between zvati and nazvati here?

Yes—aspect:

zvati (imperfective): focuses on the activity; can suggest trying/repeated calling. Zvat ću je sutra can imply “I’ll be calling/trying to call her tomorrow.”
nazvati (perfective): one completed act. Nazvat ću je sutra is often preferred for “I’ll (successfully) call her once tomorrow.”

What about pozvati?

pozvati usually means “to invite” or “to call (someone to come).” Pozvat ću je sutra is typically “I’ll invite her tomorrow,” not necessarily “phone her.” For unambiguous “phone,” use nazvati or verbs like telefonirati.

Why is it accusative ju/je and not dative?

With zvati/nazvati meaning “to call (phone) someone,” Croatian uses a direct object in the accusative: zvati koga. Hence ju/je. Verbs like telefonirati or javiti se take the dative: Telefonirat ću joj sutra. / Javit ću joj se sutra.

How do I make it negative?

Use neću (ne + ću fused) and keep the clitic pronoun close:

Neću je/ju zvati sutra.
With perfective: Neću je/ju nazvati sutra. Writing ne ću is archaic; modern standard is neću.

Can I add the subject ja?

Yes, for emphasis or contrast:

Ja ću je/ju zvati sutra. Normally it’s omitted because ću already shows 1st person singular.

Can I start the sentence with ju?

No. Unstressed clitics like ju/je/ću cannot begin a sentence. For fronted emphasis, use the stressed form:

Nju ću zvati sutra. Starting with Ju ću... is ungrammatical.

How do I pronounce the tricky bits?

ću: [tɕu], like a softer “chew”; it’s the palatal ć, not the harder č.
zv in zvat: both sounds pronounced, z as in “zoo”.
r in sutra: trilled/flapped.
Vowels are pure: u as in “boot”, a as in “father”.

Is Zvaću je sutra acceptable?

Not in standard Croatian. That synthetic future (zvaću) is standard in Serbian. In Croatian, use the periphrastic future:

Zvat ću je/ju sutra.
or Ja ću je/ju zvati sutra.

Can I put the object after the infinitive, like ću zvati ju?

Best avoid it. The clitic pronoun prefers to sit early in the clitic cluster:

Natural: ću je/ju zvati or Zvat ću je/ju.
ću zvati ju sounds odd/unidiomatic.

Can I replace ju with a name? What happens to the name?

Yes; use the accusative:

Zvat ću Anu sutra. (Ana → Anu)
Zvat ću Mariju sutra. (Marija → Mariju) For a masculine name: Zvat ću Marka sutra. (Marko → Marka)

Is zvati irregular? Why is the present zovem?

Yes, the stem alternates:

Infinitive: zvati
Present: (ja) zovem, (ti) zoveš, (on/ona) zove, etc.
Past participle: zvao/zvala So future: Zvat ću je/ju sutra, but present: Zovem je/ju svaki dan.

Where can sutra go? Does placement change meaning?

You can place sutra at the beginning, middle, or end:

Sutra ću je/ju zvati.
Zvat ću je/ju sutra.
Ja ću je/ju sutra zvati. Placement mainly affects emphasis/information flow, not the core meaning.