Στο τετράδιό μου γράφω και ένα συνώνυμο για κάθε καινούρια λέξη.

Στο is a contraction of σε + το.

• σε = in / at / to, depending on context
• το = the for a neuter singular noun

So:

• στο τετράδιό μου = in my notebook

In this sentence, στο means in.

This is because μου is an enclitic word, meaning it is unstressed and leans on the word before it.

The noun τετράδιο is stressed on the third syllable from the end. When a word like that is followed by an enclitic such as μου, Greek often adds an extra written accent:

• τετράδιο
• τετράδιό μου

So τετράδιό μου means my notebook.

This extra accent does not mean the word suddenly becomes a different word. It is mainly a spelling rule connected with stress.

In Greek, possessive words like μου (my), σου (your), του/της (his/her) usually come after the noun:

• το βιβλίο μου = my book
• η τσάντα σου = your bag
• το τετράδιό μου = my notebook

This is the normal Greek pattern.

Because Greek often omits subject pronouns when they are already clear from the verb.

Here the verb is γράφω, which means I write. The ending -ω shows the subject is I, so Greek does not need to say εγώ unless the speaker wants emphasis.

So:

• γράφω = I write
• εγώ γράφω = I write with extra emphasis, like I am the one who writes

γράφω means I write or I am writing, depending on context.

It is:

• 1st person singular
• present tense
• active voice

Greek present tense can often translate into English as either:

• I write
• I am writing

In this sentence, it is a general habit, so I write fits well.

Here και most naturally means also / too.

So:

• γράφω και ένα συνώνυμο = I also write a synonym

Greek και can mean:

• and
• also / too
• sometimes even, depending on context

In this sentence, it adds the idea that this is one more thing the speaker does in the notebook.

Συνώνυμο is a neuter singular noun, so it takes the neuter singular indefinite article ένα.

• ένα = a / an / one
• συνώνυμο = synonym

So:

• ένα συνώνυμο = a synonym or one synonym

In this sentence, it can sound slightly like one synonym for each new word, not just any number of synonyms.

Για κάθε means for each or for every.

• για = for
• κάθε = each / every

A very important point: κάθε is followed by a singular noun in Greek.

So:

• για κάθε καινούρια λέξη = for every new word

Even though the meaning is general and refers to many words, the noun stays singular after κάθε.

Καινούρια means new, and here it agrees with λέξη.

Since λέξη is:

• feminine
• singular
• in the form required by για κάθε

the adjective must match it:

• καινούρια λέξη = new word

You may also see καινούργια. Both καινούρια and καινούργια are common forms meaning new.

No. Στο τετράδιό μου is a prepositional phrase, not the direct object.

It tells you where the writing happens:

• γράφω = I write
• στο τετράδιό μου = in my notebook

The thing being written is:

• ένα συνώνυμο

So the structure is roughly:

• In my notebook = location
• I write = verb
• also a synonym = object
• for every new word = purpose/reference

Yes. Greek word order is often more flexible than English word order.

This sentence begins with Στο τετράδιό μου to highlight the place:

• Στο τετράδιό μου γράφω και ένα συνώνυμο για κάθε καινούρια λέξη.

But other orders are possible, for example:

• Γράφω και ένα συνώνυμο για κάθε καινούρια λέξη στο τετράδιό μου.

That said, the original version sounds natural and gives a nice emphasis to in my notebook at the beginning.