Learn Hardcore Greek: Σήμερα είμαι ιδιαίτερα κουρασμένος.

κουρασμένος – masculine singular (a man speaking)
κουρασμένη – feminine singular (a woman speaking)
κουρασμένο – neuter singular (for a neuter noun, e.g. a child: το παιδί είναι κουρασμένο)

So:

A man would say: Σήμερα είμαι ιδιαίτερα κουρασμένος.
A woman would say: Σήμερα είμαι ιδιαίτερα κουρασμένη.

What is the role of ιδιαίτερα here? Is it an adjective or an adverb?

ιδιαίτερα is an adverb here. It modifies the adjective κουρασμένος and means something like especially / particularly / very.

So:

ιδιαίτερα κουρασμένος = particularly tired / especially tired

If it were an adjective, you would see the form ιδιαίτερος, -η, -ο, agreeing with a noun, e.g.:

ένα ιδιαίτερο μάθημα – a special/extra class (adjective)

Can I replace ιδιαίτερα with πολύ? What’s the difference?

Yes, you can say:

Σήμερα είμαι πολύ κουρασμένος. – Today I am very tired.

πολύ = very, common, neutral, used all the time.
ιδιαίτερα = especially / particularly, a bit more emphatic or “refined,” sometimes sounds slightly more formal or expressive.

Both are correct; πολύ is more common in everyday speech.

Why does Σήμερα come at the beginning? Can it go somewhere else?

Σήμερα means today and it can move around in the sentence quite freely. All of these are grammatical:

Σήμερα είμαι ιδιαίτερα κουρασμένος.
Είμαι σήμερα ιδιαίτερα κουρασμένος.
Είμαι ιδιαίτερα κουρασμένος σήμερα.

The most natural in neutral speech is with Σήμερα at the beginning, because Greek often places time expressions first. Moving it to the end (…κουρασμένος σήμερα) puts a little extra emphasis on today.

How do you pronounce the sentence Σήμερα είμαι ιδιαίτερα κουρασμένος?

Approximate pronunciation (stress marked with capital letters):

ΣΗμερα – SEE-meh-rah
ΕΙμαι – EE-meh
ιδΙαιτερα – ee-THEE-eh-te-rah (the δ is like the th in this)
κουρασΜΕνος – koo-ras-ME-nos

So the whole sentence: SEE-meh-rah EE-meh ee-THEE-eh-te-rah koo-ras-ME-nos

What do the accent marks (´) indicate in words like σήμερα, είμαι, ιδιαίτερα, κουρασμένος?

In modern Greek, each word has one main stressed syllable, and the accent mark (´) shows where the stress falls.

σήμερα – ΣΗ-με-ρα (stress on the first syllable)
είμαι – ΕΙ-μαι (stress on the first)
ιδιαίτερα – ιδ-Ι-αι-τε-ρα (stress on the second)
κουρασμένος – κου-ρα-ΣΜΕ-νος (stress on the third)

The accent mark does not change the vowel quality; it only marks stress.

Is the word order the same as in English, or is Greek more flexible?

Greek word order is more flexible than English. The basic pattern here is similar:

Σήμερα (Today) – time expression
είμαι (I am) – verb
ιδιαίτερα (particularly) – adverb
κουρασμένος (tired) – adjective/predicate

However, because Greek uses verb endings to show the subject, you can move parts around more freely without changing the core meaning, especially for emphasis. For example:

Ιδιαίτερα κουρασμένος είμαι σήμερα. – Emphasizes particularly tired.

Why don’t we need a pronoun like εγώ (“I”) before είμαι?

In Greek, the verb ending shows the subject, so the subject pronoun is usually omitted unless you want to emphasize it.

είμαι already means I am.
εγώ είμαι would sound like I am (as opposed to someone else), adding emphasis to I.

So:

Neutral: Σήμερα είμαι ιδιαίτερα κουρασμένος.
Emphatic: Σήμερα εγώ είμαι ιδιαίτερα κουρασμένος. (I am the one who is especially tired today.)

How would I say “Today we are particularly tired” instead?

Change the verb to the 1st person plural and keep the adjective in the plural masculine (or mixed group) form:

Σήμερα είμαστε ιδιαίτερα κουρασμένοι.

Forms:

είμαστε – we are
κουρασμένοι – tired (masculine plural; used for all-male or mixed groups)
For an all-female group:
Σήμερα είμαστε ιδιαίτερα κουρασμένες.

Is κουρασμένος always used with είμαι, or can it be used in other ways?

κουρασμένος is an adjective meaning tired and is often used with είμαι, but not only:

Είμαι κουρασμένος. – I am tired.
Φαίνεσαι κουρασμένος. – You seem tired.
Έμεινε κουρασμένος όλη μέρα. – He remained tired all day.

It can also be used attributively before a noun:

ο κουρασμένος άντρας – the tired man
η κουρασμένη γυναίκα – the tired woman

What is the difference between σήμερα (today) and τώρα (now)? Could I say Τώρα είμαι ιδιαίτερα κουρασμένος?

σήμερα = today (the whole day)
τώρα = now (this moment)

Yes, you can say:

Τώρα είμαι ιδιαίτερα κουρασμένος. – Right now I am particularly tired.

This focuses on the present moment, not the whole day. Both are correct, but they express slightly different time frames.