Στο τέλος της μέρας, θέλω μόνο να βγάλω τα παπούτσια μου και να χαλαρώσω.

Στο is a contraction of σε + το.

• σε = in, at, to
• το = the

So στο τέλος literally means at the end. This contraction is extremely common in Greek:

• στο σπίτι = at home / to the house
• στο γραφείο = at the office

Because της μέρας is in the genitive case, and Greek uses the genitive after τέλος to mean the end of something.

So:

• το τέλος = the end
• της μέρας = of the day

Together: το τέλος της μέρας = the end of the day

This is similar to English expressions like the end of the day or the day’s end.

They are forms of the same word:

• η μέρα = the day
• η ημέρα = the day

Μέρα is the shorter, very common everyday form. Ημέρα is more formal or more careful speech/writing.

So:

• στο τέλος της μέρας = natural, everyday
• στο τέλος της ημέρας = a bit more formal

Both are correct.

Modern Greek does not normally use an infinitive the way English does.

So instead of saying something like I want take off, Greek uses:

• θέλω να βγάλω
• literally: I want that I take off
• naturally: I want to take off

The particle να introduces this kind of verb form, often called the subjunctive in Greek grammar.

This pattern is very common:

• θέλω να φύγω = I want to leave
• μπορώ να έρθω = I can come
• πρέπει να δουλέψω = I must work

This is about aspect, which is very important in Greek.

After να, Greek often chooses between:

• a perfective / aorist form: βγάλω, χαλαρώσω
• an imperfective / present form: βγάζω, χαλαρώνω

Here, βγάλω and χαλαρώσω present the actions as whole, complete actions:

• να βγάλω τα παπούτσια μου = to take off my shoes
• να χαλαρώσω = to relax, to unwind

If you used the present-type forms, the meaning would feel more ongoing, repeated, or habitual:

• θέλω να βγάζω τα παπούτσια μου = I want to be taking off my shoes / I want to take off my shoes regularly
• θέλω να χαλαρώνω = I want to relax habitually / I want to be relaxing

So the forms in the sentence are the natural ones for one occasion at the end of the day.

Greek normally repeats να before each coordinated verb:

• θέλω μόνο να βγάλω τα παπούτσια μου και να χαλαρώσω

This is the most standard and natural structure.

English says to take off my shoes and relax, with to only once, but Greek usually prefers:

• να βγάλω ... και να χαλαρώσω

As a learner, repeating να in this pattern is the safest and most natural choice.

The basic verb is βγάζω, which can mean things like:

• take out
• remove
• bring out
• produce

When used with clothing, shoes, glasses, etc., it often means take off:

• βγάζω τα παπούτσια μου = I take off my shoes
• βγάζω το παλτό μου = I take off my coat
• βγάζω τα γυαλιά μου = I take off my glasses

So in this sentence, να βγάλω τα παπούτσια μου is exactly the natural Greek way to say to take off my shoes.

Μου here means my, but in Greek possessive words of this type usually come after the noun.

So:

• τα παπούτσια μου = my shoes
• literally: the shoes my

This is a very common Greek structure:

• το σπίτι μου = my house
• η αδερφή μου = my sister
• οι φίλοι μου = my friends

So even though English puts my before the noun, Greek usually puts μου after it.

It is in the accusative case because it is the direct object of βγάλω:

• να βγάλω τι; = to take off what?
• τα παπούτσια μου = my shoes

A useful detail: with many neuter plural nouns, the nominative and accusative forms look the same.

So:

• nominative: τα παπούτσια
• accusative: τα παπούτσια

That is why there is no visible change here.

Μόνο means only or just.

In this sentence:

• θέλω μόνο να βγάλω τα παπούτσια μου και να χαλαρώσω
• I only / just want to take off my shoes and relax

Here μόνο limits the speaker’s wish: that is all they want.

Its position matters. Compare:

• Θέλω μόνο να βγάλω τα παπούτσια μου και να χαλαρώσω
  = I just want to take off my shoes and relax

• Θέλω να βγάλω μόνο τα παπούτσια μου
  = I want to take off only my shoes

So the sentence’s word order is chosen to mean That’s all I want to do.

It can be both.

In this sentence, it is most naturally understood literally:

• At the end of the day, I just want to take off my shoes and relax

That means after the day is over, when the speaker gets home or finishes working.

But Greek can also use this phrase more idiomatically, like English at the end of the day, meaning when all is said and done or ultimately.

A slightly more formal version of that idiomatic expression is often:

• στο τέλος της ημέρας

Greek generally uses articles much more regularly than English.

So Greek says:

• της μέρας = of the day

not just of day.

This is normal Greek grammar. Articles are usually kept with nouns unless there is a specific reason to omit them.

It is pronounced roughly VGA-lo, with the stress on the first syllable:

• βγά-λω

More precisely, the first sound is not exactly English g. The γ here is the Greek sound that often appears as a voiced throat/back sound. But for a learner, thinking VGA-lo is a good start.

The important point is that βγ comes together at the beginning, so do not insert an extra vowel:

• not be-ga-lo
• more like vga-lo