Κάθε Τετάρτη κάνω μάθημα ελληνικών στο φροντιστήριο.

In Greek, subject pronouns (εγώ = I, εσύ = you, etc.) are usually left out, because the verb ending already shows who the subject is.

• κάνω = “I do / I make / I have (a lesson)”
  
  • -ω ending → 1st person singular → I

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.”).

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)

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”)

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.

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.

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.

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

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.

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)

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.

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.)

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.

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.

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).

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.