Τον Σεπτέμβριο αρχίζει πάλι το πανεπιστήμιο, κι εγώ βάζω τα βιβλία μου σε μια καινούρια τσάντα.

Because Greek often uses the accusative for expressions of time such as months. The dictionary form is ο Σεπτέμβριος, but in this sentence it becomes τον Σεπτέμβριο, meaning in September or during September.

This same pattern appears with other months too:

• τον Οκτώβριο
• τον Νοέμβριο
• τον Δεκέμβριο

So Τον Σεπτέμβριο is a normal time expression, not the subject of the sentence.

Because the subject is το πανεπιστήμιο, which is third person singular: it begins.

• αρχίζω = I begin / I start
• αρχίζει = he/she/it begins or it starts

Greek often puts the subject after the verb, so even though το πανεπιστήμιο comes later, it is still the thing doing the action.

Greek word order is more flexible than English word order. In English, you usually expect the university starts, but in Greek αρχίζει το πανεπιστήμιο is perfectly natural.

Putting the verb first can sound very normal in narration. The meaning does not change: το πανεπιστήμιο is still the subject.

So:

• Το πανεπιστήμιο αρχίζει = normal
• Αρχίζει το πανεπιστήμιο = also normal

The second version may sound a little more like the university is starting or may simply fit the rhythm of the sentence better.

Πάλι usually means again. In this sentence it means that the university starts again, probably after a break.

Depending on context, πάλι can sometimes also feel like back or once more, but again is the clearest basic meaning here.

Greek uses the definite article much more often than English. So το πανεπιστήμιο is very natural even in cases where English might simply say university.

Greek often says:

• πάω στο πανεπιστήμιο = I go to university / the university
• αρχίζει το πανεπιστήμιο = university starts / the university starts

So the article does not always sound as specific in Greek as the does in English.

Κι is a shortened form of και, and it means and.

In front of a vowel sound, Greek very often uses κι instead of και, especially in natural speech and normal writing:

• κι εγώ = and I
• κι αυτός = and he
• κι εκεί = and there

So κι εγώ and και εγώ both mean the same thing, but κι εγώ sounds smoother here.

Yes, Greek could leave it out, because the verb form βάζω already shows that the subject is I.

But εγώ is included for emphasis or contrast. It gives a feeling like:

• and I
• as for me
• and I, meanwhile

So κι εγώ βάζω... is more expressive than just κι βάζω....

In Greek, possessive words like μου, σου, του, της usually come after the noun.

So:

• τα βιβλία μου = my books
• η τσάντα μου = my bag
• ο φίλος μου = my friend

This is one of the most basic Greek word-order patterns for possession:

• article + noun + possessive pronoun

So τα βιβλία μου is the normal way to say my books.

There are a few things happening there:

1. σε means in, into, to, depending on context.
   With βάζω it often means put into or put in.

2. μια means a / one and is the feminine singular form of the indefinite article.

3. καινούρια agrees with τσάντα, which is feminine singular.
   So both are feminine singular:

   • μια
   • καινούρια
   • τσάντα

Also, after σε, Greek uses the accusative. In this phrase, the forms happen to look the same as the nominative because τσάντα is a feminine noun of this type.

So the whole phrase means into a new bag or in a new bag.

Yes, βάζω is present tense. Greek present tense can be used for:

• a habitual action
• a repeated situation
• a vivid description of what happens at a certain time

So the sentence can sound like a general, recurring situation: when September comes and university starts again, I put my books in a new bag.

If the speaker wanted to make the future meaning more explicit, Greek could also use θα:

• θα αρχίσει
• θα βάλω

But the plain present is still possible when talking about regular or expected events.

Yes. Greek allows a lot of flexibility in word order, especially for emphasis.

For example, these are all possible:

• Τον Σεπτέμβριο αρχίζει πάλι το πανεπιστήμιο...
• Το πανεπιστήμιο αρχίζει πάλι τον Σεπτέμβριο...
• Κι εγώ βάζω τα βιβλία μου σε μια καινούρια τσάντα.
• Τα βιβλία μου βάζω σε μια καινούρια τσάντα.

The basic meaning stays similar, but the emphasis changes. In your sentence:

• Τον Σεπτέμβριο is placed first to set the time frame
• κι εγώ is placed early to emphasize I in contrast with the first clause

So the order is not random; it helps shape the focus of the sentence.