Learn Hardcore Greek: Σήμερα κάθομαι δίπλα στη φίλη μου στο πανεπιστήμιο. - Today I am sitting next to my friend at the university.

Greek usually leaves out subject pronouns like εγώ (I) because the verb ending already shows who the subject is.
In κάθομαι, the ending -ομαι tells us it is 1st person singular → I.
You can add εγώ (Εγώ σήμερα κάθομαι…) for emphasis (e.g. I today am sitting…), but it is not needed for normal sentences.

What exactly does κάθομαι mean, and why does it end in -ομαι?

κάθομαι is a so‑called “deponent” verb: it has middle/passive endings (-ομαι) but an active meaning.
Its main meanings are:

I am sitting / I sit (state: being in a seated position)
I sit down (action: moving into a seated position, depending on context)

In English we distinguish I sit vs I am sitting, but Greek uses the same present form κάθομαι for both.
There is no separate κάθω in modern standard Greek; κάθομαι is the normal dictionary form.

What does δίπλα στη mean, and why do we need both δίπλα and σε?

δίπλα by itself means “next to, beside” in a general sense.
To say next to someone/something, Greek normally uses:

δίπλα σε + noun → δίπλα στη φίλη μου = next to my friend

Here σε contracts with τη (the feminine article) to form στη:

σε + τη φίλη → στη φίλη

So δίπλα στη φίλη μου literally is “beside to-the friend my”, which in English is just “next to my friend”.

What is στη exactly? Why not just σε τη φίλη μου?

Spoken Greek very often contracts σε + definite article:

σε + τον → στον
σε + τη → στη
σε + το → στο
σε + τις → στις
σε + τα → στα

So in normal modern Greek you almost always say στη φίλη μου, not σε τη φίλη μου.
These contracted forms (στον, στη, στο, στις, στα) are completely standard and are what you should learn and use.

Why is it στη φίλη and not στην φίλη?

You will see both στη and στην in Greek. The final -ν is called νι-εφελκυστικό (movable ν) and is:

kept before vowels and certain consonants (π, τ, κ, ξ, ψ, μπ, ντ, γκ, τσ, τζ, and double consonants), e.g. στην πόλη, στην Ελλάδα
optional or usually dropped before other consonants.

Since φ is not in that special list, it is very common and fully correct to write στη φίλη μου.
Some speakers might still say/write στην φίλη μου, but στη φίλη μου matches the standard spelling rule.

Why does Greek say στη φίλη μου (with an article) when English says “my friend” without “the”?

In Greek, possessives like μου usually come after the noun and they do not replace the article:

η φίλη μου = the friend my → my friend
το σπίτι μου = the house my → my house

So στη φίλη μου literally is “to the friend my”, but it simply means “to my friend / next to my friend”.
Greek uses the definite article much more often than English; having both article + possessive is normal.

Why is φίλη (female friend) used here instead of φίλος?

Greek nouns have grammatical gender:

ο φίλος = male friend
η φίλη = female friend

Because we are talking about a female friend, the noun is η φίλη, and the article in the phrase with σε must match that gender:

σε + τη (φίλη) → στη φίλη

If it were a male friend, you’d say:

δίπλα στον φίλο μου = next to my (male) friend

Why is it στο πανεπιστήμιο and not something like στη πανεπιστήμιο?

πανεπιστήμιο is a neuter noun: το πανεπιστήμιο.
With the preposition σε, the neuter article το becomes στο:

σε + το πανεπιστήμιο → στο πανεπιστήμιο

στη is only used with feminine nouns (from τη).
So we have:

στη φίλη (feminine)
στο πανεπιστήμιο (neuter)

Why does σε take the accusative case (τη φίλη, το πανεπιστήμιο)?

In modern Greek, σε is a preposition that always takes the accusative case.
That is why we say:

στη φίλη (accusative of η φίλη)
στο πανεπιστήμιο (accusative of το πανεπιστήμιο)

You do not need to change the preposition; you just learn that σε + noun → noun in the accusative.

Why is it στο πανεπιστήμιο (“at the university”) when English usually just says “at university” with no article?

Greek uses the definite article much more frequently than English, especially with places and institutions:

στο σχολείο = (at) school
στο γραφείο = (at the) office
στο πανεπιστήμιο = (at) university

English very often leaves the article out in these expressions, but Greek normally keeps it.
So στο πανεπιστήμιο is the natural Greek way to say at university.

Could the word order be different? For example, can I say Κάθομαι σήμερα δίπλα στη φίλη μου στο πανεπιστήμιο?

Yes. Greek word order is quite flexible, especially with adverbs of time and place.
All of these are grammatically fine (same basic meaning, slightly different emphasis):

Σήμερα κάθομαι δίπλα στη φίλη μου στο πανεπιστήμιο.
Κάθομαι σήμερα δίπλα στη φίλη μου στο πανεπιστήμιο.
Σήμερα, στο πανεπιστήμιο, κάθομαι δίπλα στη φίλη μου.

Putting σήμερα at the beginning is very common, because it nicely sets the time frame of the sentence.

Does κάθομαι here mean “I sit” or “I am sitting”? Is there a difference in Greek?

Greek present tense normally covers both English simple present and present continuous.
So Σήμερα κάθομαι δίπλα στη φίλη μου can be:

Today I am sitting next to my friend (what is happening today)
In some contexts, also Today I sit next to my friend (habit, schedule)

The context decides how you translate it into English, but the Greek form κάθομαι stays the same.