Βάζω τα πιάτα στην πιατοθήκη όταν τελειώνω το πλύσιμο.

Greek often drops the subject pronoun when it is clear from the verb ending.
Βάζω already means I put / I am putting, and τελειώνω already means I finish / I am finishing.

So εγώ is not needed unless you want extra emphasis or contrast, for example:

Εγώ βάζω τα πιάτα... = I put the dishes...
implying maybe someone else does something different.

It can mean either one, depending on context.

The Greek present tense often covers both:

• simple present: I put
• present continuous: I am putting

In this sentence, because of όταν and the general everyday context, it sounds habitual: I put the dishes in the dish rack/cupboard when I finish the washing up
or more naturally in English, Whenever I finish washing up, I put the dishes away.

Στην is a very common contraction of σε + την.

• σε = in, at, on, to
• την = the for a feminine noun in the accusative

So:

σε την πιατοθήκη → στην πιατοθήκη

This contraction is standard and extremely common. Similar forms are:

• στο = σε το
• στον = σε τον
• στις = σε τις

Yes. Both are in the accusative, but for different reasons.

• τα πιάτα is the direct object of βάζω: what am I putting? The dishes.
• την πιατοθήκη is after σε / στην, and Greek normally uses the accusative after this preposition.

A useful detail:

• For many neuter nouns like πιάτο, the nominative and accusative look the same in the plural: τα πιάτα
• For the feminine noun πιατοθήκη, the article changes:
  η πιατοθήκη = nominative
  την πιατοθήκη = accusative

Όταν means when or whenever.

In this sentence it introduces a time clause: όταν τελειώνω το πλύσιμο = when/whenever I finish the washing up

Because the whole sentence describes a repeated, normal action, όταν is best understood as whenever in natural English.

This sentence describes a habitual/repeated action, so Greek uses the present indicative:

όταν τελειώνω το πλύσιμο = when/whenever I finish the washing up

If you were talking about one future occasion, Greek would usually use the other form:

Όταν τελειώσω το πλύσιμο, θα βάλω τα πιάτα στην πιατοθήκη.
= When I finish the washing up, I will put the dishes in the dish rack/cupboard.

So:

• τελειώνω = repeated/habitual I finish
• τελειώσω = one completed future event I finish

Το πλύσιμο is a noun, literally the washing.

It comes from the verb πλένω = I wash. Greek very often uses this kind of noun where English might use:

• a gerund, like washing
• or a phrase like the washing up

So το πλύσιμο here means something like:

• the washing up
• the dishwashing
• the washing

depending on context.

This is very natural Greek.

Πιατοθήκη is a compound noun:

• πιάτο = plate / dish
• θήκη = holder, case, container, storage place

So the literal idea is dish-holder or plate-storage place.

Depending on the home and context, it can mean:

• dish rack
• dish cupboard
• plate rack

So the exact English word can vary a bit, but the Greek word is perfectly normal.

It can cover several of those meanings. Greek σε is broader than any single English preposition.

Depending on context, σε can mean:

• in
• at
• on
• to

So στην πιατοθήκη could be translated as in the dish cupboard or on/in the dish rack, depending on what kind of πιατοθήκη you imagine.

This is very common in Greek: one preposition often maps to several English ones.

Yes. Greek word order is more flexible than English, because the verb endings and articles help show the grammar.

The given sentence is perfectly natural: Βάζω τα πιάτα στην πιατοθήκη όταν τελειώνω το πλύσιμο.

But you could also say: Όταν τελειώνω το πλύσιμο, βάζω τα πιάτα στην πιατοθήκη.

Both are correct. The second version brings the when-clause to the front, a bit like English When I finish washing up, I put the dishes away.

Modern Greek usually does not use an infinitive the way English does. English can say:

• after finishing
• when I finish

Greek normally prefers a full clause with a finite verb, such as:

όταν τελειώνω το πλύσιμο

So even when English uses a shorter structure, Greek often uses a more explicit verb form. That is a normal pattern in Modern Greek.

It sounds like a routine or habit.

That feeling comes mainly from:

• the present tense: βάζω, τελειώνω
• the time word όταν

So the idea is not right now I am doing this once, but rather this is what I usually do.

If Greek wanted to focus on a single future event, it would normally use forms like: Όταν τελειώσω..., θα βάλω...