Αντίο, θα σε πάρω αύριο το πρωί.

Because after θα, Greek often uses the perfective verb form when talking about a single complete future action.

• παίρνω = I take / I am taking / I take regularly
• θα πάρω = I will take / I will call / I will pick up

So in this sentence, θα σε πάρω means a one-time future action.

A useful contrast:

• Θα σε πάρω αύριο. = I’ll call you / pick you up tomorrow.
• Θα σε παίρνω κάθε μέρα. = I’ll call you every day.

The second one sounds habitual or repeated.

Θα is the Greek future particle. It does the job that will does in English.

So:

• θα πάρω = I will call / take / pick up
• θα δω = I will see
• θα φύγω = I will leave

It does not change form. You use θα with the verb to make future meaning.

σε is the weak object pronoun meaning you, and in Greek these short object pronouns usually come before the verb.

So:

• θα σε πάρω = I will call you
• σε βλέπω = I see you
• σε θέλω = I want you

English puts you after the verb, but Greek often puts this short pronoun before it.

Also, σε is for singular informal you.
If you are talking to more than one person, or using formal you, Greek uses σας:

• Θα σας πάρω αύριο. = I’ll call you tomorrow.

Yes — παίρνω can mean several things, and call is one of them.

Common meanings of παίρνω include:

• take
• get
• receive
• pick up
• catch
• call someone on the phone

So θα σε πάρω can mean:

• I’ll call you
• I’ll pick you up

The exact meaning depends on the context. Greek often leaves this to context, whereas English usually chooses a more specific verb.

You usually know from the situation.

For example:

• In a phone conversation, θα σε πάρω αύριο το πρωί very naturally means I’ll call you tomorrow morning.
• If two people are arranging transport, it could mean I’ll pick you up tomorrow morning.

Greek speakers do not always add extra words to make it explicit, because the context is usually enough.

Because that is the normal Greek way to say tomorrow morning.

Word by word:

• αύριο = tomorrow
• το πρωί = the morning / in the morning

So literally it looks like tomorrow the morning, but the natural English translation is tomorrow morning.

Greek very often uses the article with parts of the day:

• το πρωί = in the morning
• το μεσημέρι = at noon / midday
• το απόγευμα = in the afternoon
• το βράδυ = in the evening / at night

Yes, Greek word order is more flexible than English.

These are all possible, though they may emphasize different things:

• Θα σε πάρω αύριο το πρωί.
• Αύριο το πρωί θα σε πάρω.

Both are natural. The second one puts more focus on when.

But the short pronoun σε normally stays before the verb, so:

• θα σε πάρω = natural
• θα πάρω σε = not natural in ordinary Greek

Αντίο means goodbye or bye.

It is a parting word used when ending a conversation. In this sentence, it sets the tone:

• Αντίο, θα σε πάρω αύριο το πρωί.
  = Bye, I’ll call you tomorrow morning.

In everyday speech, Greek speakers also often use γεια as both hello and bye, but αντίο is a clear goodbye.

A rough pronunciation is:

an-DEE-o, tha se PA-ro AV-rio to pro-EE

A few notes:

• Αντίο has the stress on -τί-
• πάρω has the stress on πά-
• αύριο is usually pronounced roughly AV-rio
• πρωί has the stress on the last syllable: pro-EE

Because Greek often expresses time phrases without a direct equivalent of English in.

So Greek says:

• το πρωί = in the morning
• το βράδυ = in the evening / at night

This is very normal. You do not need a separate preposition there.

It is informal, because of σε, which means you singular informal.

If you wanted to say the same thing to someone formally, or to more than one person, you would use σας:

• Αντίο, θα σας πάρω αύριο το πρωί.

So the sentence you have is the kind of thing you would say to a friend, relative, partner, or someone you address with informal you.