Την Πρωτοχρονιά τρώμε πρωινό μαζί στο σπίτι.

Because Greek often uses the accusative for expressions of time meaning when something happens.

• η Πρωτοχρονιά = New Year’s Day / New Year as the basic dictionary form
• την Πρωτοχρονιά = on New Year’s Day

So the article changes from η to την because the phrase is functioning like a time expression.

Here, Πρωτοχρονιά means New Year’s Day or the New Year holiday, not the whole new year as a 12-month period.

So Την Πρωτοχρονιά is best understood as On New Year’s Day.

Greek often leaves out subject pronouns because the verb ending already tells you who the subject is.

• τρώμε = we eat

So εμείς is not necessary. You would only add εμείς for emphasis or contrast:

• Εμείς τρώμε πρωινό μαζί στο σπίτι. = We eat breakfast together at home.

Τρώμε is the 1st person plural present tense of τρώω, which means to eat.

• τρώω = I eat
• τρώμε = we eat

The ending -με is a very useful clue that the subject is we.

That is just the normal Greek way to express it. Greek commonly uses eat with meals:

• τρώω πρωινό = eat breakfast
• τρώω μεσημεριανό = eat lunch
• τρώω βραδινό = eat dinner

So even though English prefers have breakfast, Greek naturally uses eat breakfast.

Because meal words are often used without an article when talking about the activity in a general way.

• τρώμε πρωινό = we eat breakfast

If you wanted to talk about a specific breakfast, you could use the article:

• Τρώμε το πρωινό που ετοίμασες. = We’re eating the breakfast you prepared.

So in your sentence, no article is the normal choice.

Here it is a noun meaning breakfast.

But the same form can also function as an adjective meaning morning in other contexts. In this sentence, though, it is clearly a noun because it is the thing being eaten.

Μαζί means together.

In this sentence, it tells you how the action happens: the people eat breakfast together.

Its position is fairly flexible in Greek, although some placements sound more natural than others. For example:

• Την Πρωτοχρονιά τρώμε πρωινό μαζί στο σπίτι.
• Την Πρωτοχρονιά τρώμε μαζί πρωινό στο σπίτι.

Both are possible, but the original sentence sounds very natural.

Στο is the contracted form of σε το.

• σε = in, at, to
• το σπίτι = the house / home
• στο σπίτι = in the house / at home

In this sentence, στο σπίτι is best translated as at home.

Also, σπίτι is a neuter noun, and after σε it is in the accusative, but for this noun the nominative and accusative look the same: σπίτι.

Because Greek and English do not use articles in exactly the same way.

Greek very naturally says στο σπίτι for at home, even though English does not normally say at the home in this meaning. So this is something best learned as a natural Greek expression rather than translated word-for-word.

No. Greek word order is more flexible than English word order.

The sentence you have is a very natural order:

• Την Πρωτοχρονιά = time
• τρώμε πρωινό = main action
• μαζί = together
• στο σπίτι = place

But other orders are possible if you want a different emphasis, for example:

• Στο σπίτι τρώμε πρωινό μαζί την Πρωτοχρονιά.
• Την Πρωτοχρονιά μαζί τρώμε πρωινό στο σπίτι.

They do not all sound equally neutral, but they are understandable.

Grammatically, τρώμε is present tense.

In context, it most naturally sounds like a habit or custom:

• On New Year’s Day, we eat breakfast together at home.

Greek, like English, can sometimes use the present for scheduled or regular future events, but in this sentence the most likely meaning is a repeated tradition.

If you wanted to make the future very explicit, you could say:

• Θα φάμε πρωινό μαζί στο σπίτι. = We will eat breakfast together at home.

Because it is the name of a holiday, and holiday names are often capitalized in Greek.

So Πρωτοχρονιά with a capital letter is normal. You may sometimes see lowercase in less formal writing, but the capitalized form is very common.