Learn Hardcore Greek: Κάθε Τετάρτη κάνω μάθημα ελληνικών στο φροντιστήριο. - Every Wednesday I have a Greek lesson at the language school.

So κάνω μάθημα already means “I have a lesson”.
You would only add εγώ for emphasis or contrast, e.g. Εγώ κάνω μάθημα, όχι εσύ (“I have a lesson, not you.”).

Why is Τετάρτη (Wednesday) singular when English would say “on Wednesdays”?

Greek uses the singular with κάθε (“every/each”):

κάθε Τετάρτη = “every Wednesday” / “on Wednesdays”
literally: “each Wednesday”

With κάθε, the noun is always singular.
If you drop κάθε, you can say:

τις Τετάρτες = “(on) Wednesdays” (plural with article)

Why is there no article before Τετάρτη? Why not κάθε την Τετάρτη?

The word κάθε (“every”) does not take an article before the noun:

κάθε Τετάρτη ✔
κάθε την Τετάρτη ✘ (wrong)
κάθε μέρα = every day
κάθε βράδυ = every evening

If you use the plural instead, then you use the article:

τις Τετάρτες (“on Wednesdays”)

Are days of the week always capitalized in Greek, like Τετάρτη here?

In modern Greek, the standard rule is:

Days of the week are normally written with a lowercase:
- τετάρτη, δευτέρα, τρίτη, etc.
They are capitalized only at the beginning of a sentence or in titles.

In your sentence Κάθε Τετάρτη…, it’s at the start of the sentence, so capitalization is normal.
If it were in the middle of a sentence, you would usually see κάθε τετάρτη.

What exactly does κάθε mean here, and how is κάθε Τετάρτη different from τις Τετάρτες?

κάθε means “every / each”:

κάθε Τετάρτη = “every Wednesday” (focus on regular repetition)
τις Τετάρτες = “(on) Wednesdays” (more neutral plural, often descriptive)

In practice they overlap a lot:

Κάθε Τετάρτη κάνω μάθημα.
Τις Τετάρτες κάνω μάθημα.

Both can be understood as “I have class on Wednesdays,” but κάθε Τετάρτη sounds a bit more like a schedule or routine.

Why does Greek say κάνω μάθημα instead of something like “I have a lesson”?

Greek often uses κάνω (“I do / I make”) in expressions where English uses “have”:

κάνω μάθημα = I have a lesson / I am in class
κάνω διάλειμμα = I take a break
κάνω μπάνιο = I take a bath/shower

So κάνω μάθημα ελληνικών is the natural way to say “I have a Greek lesson / Greek class.”

You can also say έχω μάθημα ελληνικών, and it’s correct, but κάνω μάθημα is extremely common in school/tutoring contexts.

What is the literal meaning of μάθημα ελληνικών? Why is ελληνικών in that form?

Literally:

μάθημα = “lesson, class”
ελληνικών = genitive plural of ελληνικά (“Greek” as a language)

So μάθημα ελληνικών literally means “lesson of Greek”.

Grammar:

ελληνικά (neuter plural) = Greek (the language), literally “Greek things/words”
Genitive plural: (των) ελληνικών = “of Greek”

In this phrase, the article is dropped:

μάθημα (των) ελληνικών → μάθημα ελληνικών

This genitive without article is very common to show the subject of a course:

μάθημα μαθηματικών = math class
μάθημα αγγλικών = English class
μάθημα ελληνικών = Greek class

Why is ελληνικών not capitalized even though “Greek” is a language?

In Greek, names of languages and nationalities are not capitalized, unless they start a sentence:

τα ελληνικά = Greek (language)
τα αγγλικά = English
τα γαλλικά = French

So ελληνικών in μάθημα ελληνικών is correctly written with a lowercase initial.
This is different from English, where “Greek, English, French” are capitalized.

Why is the preposition στο used? What does it consist of?

στο is a contraction of:

σε (in, at, to) + το (the, neuter singular) → στο

So:

στο φροντιστήριο = “at the tutoring school / at the cram school”

You use σε + definite article for locations:

στο σχολείο = at the school
στη δουλειά = at work
στο σπίτι = at home

If you want to say “at a language school” (not a specific one), you’d say:

σε ένα φροντιστήριο = at a (certain) cram school
(no contraction because it’s σε + ένα, not σε + το)

Could I omit the article and just say σε φροντιστήριο?

You can, but the meaning changes slightly:

στο φροντιστήριο = at the (known, specific) tutoring school
σε ένα φροντιστήριο = at a (one, not specified) tutoring school
σε φροντιστήριο = at (some) tutoring school in general, more vague/indefinite

In everyday speech, if you mean “the place where I regularly go for lessons,” στο φροντιστήριο is the most natural.

What exactly is a φροντιστήριο? How is it different from σχολείο?

φροντιστήριο is a kind of private tutoring center / cram school / language school.
In Greece it usually means:

private lessons (often afternoon/evening)
language courses, exam preparation, extra help with school subjects

σχολείο is the general word for school (primary, secondary, etc.).
So:

σχολείο = regular school
φροντιστήριο = private extra classes (languages, exams, etc.)

Can the word order of the sentence be changed? Are other orders still correct?

Yes, Greek word order is quite flexible. All of these are grammatically correct:

Κάθε Τετάρτη κάνω μάθημα ελληνικών στο φροντιστήριο. (neutral)
Κάθε Τετάρτη στο φροντιστήριο κάνω μάθημα ελληνικών. (slight emphasis on location)
Στο φροντιστήριο κάθε Τετάρτη κάνω μάθημα ελληνικών. (emphasis on “at the tutoring school”)
Μάθημα ελληνικών κάνω κάθε Τετάρτη στο φροντιστήριο. (emphasis on “Greek lesson”)

The basic information stays the same; changing the order mainly adds emphasis or focus to different parts.

Why is the verb in the simple present (κάνω) and not in some kind of “continuous” or “future” tense?

In Greek, the simple present is used for:

habits and routines
general truths
things that happen regularly

So Κάθε Τετάρτη κάνω μάθημα… naturally means:

“Every Wednesday I have class…” (habitual action)

If you wanted to express the future schedule, you could say:

Κάθε Τετάρτη θα κάνω μάθημα… = Every Wednesday I will (be) having class… (future plan)

But for your normal, ongoing routine, present tense is exactly what Greek uses.

Why is μάθημα singular in Greek, while in English we often say “I have Greek lessons”?

Greek often uses the singular with a time expression to describe a recurring activity:

Κάθε Τετάρτη κάνω μάθημα ελληνικών.
Literally: “Every Wednesday I do a Greek lesson.”
Meaning: “I have Greek lessons (every Wednesday).”

The combination of κάθε + time already shows repetition, so Greek doesn’t need the plural.
If you talk more generally about the subject, you might use the plural:

Κάνω μαθήματα ελληνικών. = I take Greek lessons (in general, not tied to a specific day).

What are the genders of μάθημα and φροντιστήριο, and does that affect the sentence?

Both μάθημα and φροντιστήριο are neuter nouns:

το μάθημα (neuter) = the lesson
το φροντιστήριο (neuter) = the cram school / language school

You see this in:

το μάθημα, το φροντιστήριο
contraction: σε + το → στο φροντιστήριο

Their neuter gender mainly affects which article and adjectives agree with them.
In your sentence, it explains why we get στο φροντιστήριο and not e.g. στη φροντιστήριο.