Θέλω να πάω στην Πάτρα το καλοκαίρι, αλλά δεν ξέρω αν θα βρω απευθείας πτήση.

Because να is needed before this kind of verb form in Greek.

• Θέλω = I want
• να πάω = to go / more literally, a form like that I go

After verbs like θέλω (I want), Greek normally uses να + verb:

• Θέλω να φύγω = I want to leave
• Θέλω να δω = I want to see

So να is very common where English uses the infinitive to.

Yes, both are related to going, but they are used a bit differently.

In this sentence, να πάω is very natural because it refers to one planned trip: going to Patras on one occasion.

• πάω often works well for a single, specific act of going
• πηγαίνω often feels more like I go / I am going / I usually go, depending on context

So:

• Θέλω να πάω στην Πάτρα = I want to go to Patras
• Πηγαίνω στην Πάτρα κάθε καλοκαίρι = I go to Patras every summer

A learner will often hear both forms, but πάω is especially common in everyday speech.

στην is a contraction of σε + την.

• σε = to / in / at
• την = the (feminine accusative singular)

So:

• σε την Πάτρα becomes στην Πάτρα

Greek uses this very often:

• στον = σε + τον
• στη / στην = σε + τη(ν)
• στο = σε + το

Since Πάτρα is a feminine noun, you get στην Πάτρα.

Greek often uses the definite article with place names, much more than English does.

So Greek naturally says:

• στην Πάτρα
• στην Αθήνα
• στο Λονδίνο

Even though English usually says just to Patras, to Athens, to London, Greek often keeps the article.

This is normal and very common.

Because Πάτρα is a feminine noun whose nominative and accusative singular look the same.

After σε / στην, Greek uses the accusative. But for many feminine nouns ending in -α, nominative and accusative are identical:

• nominative: η Πάτρα
• accusative: την Πάτρα

So the form stays Πάτρα.

Greek often uses the article in time expressions like this.

• το καλοκαίρι = in summer / in the summer
• τον χειμώνα = in winter
• την Κυριακή = on Sunday

So the article here is completely normal.
It does not necessarily mean a very specific summer; it can also just mean during the summer in a general sense.

In Greek, δεν is the usual negation for indicative verbs, and it comes right before the verb.

• ξέρω = I know
• δεν ξέρω = I don’t know

Examples:

• Δεν καταλαβαίνω = I don’t understand
• Δεν θέλω = I don’t want

So αλλά δεν ξέρω simply means but I don’t know.

Here αν means if / whether in the sense of I don’t know if/whether...

So:

• δεν ξέρω αν θα βρω... = I don’t know if / whether I’ll find...

In English, if and whether are both possible here.
Greek uses αν very naturally in this kind of clause.

It can also introduce real conditions:

• Αν έρθει, θα χαρώ. = If he comes, I’ll be happy.

So the same word can cover both ideas.

Because Greek future forms show aspect, not just tense.

• βρω is the perfective form of βρίσκω
• θα βρω suggests finding something as a single complete event
• θα βρίσκω would suggest something repeated, ongoing, or habitual, which does not fit here

In this sentence, the speaker means:

• I don’t know if I’ll manage to find a direct flight

That is one complete event, so θα βρω is the natural choice.

Compare:

• Θα βρω λύση. = I’ll find a solution.
• Θα βρίσκω χρόνο κάθε μέρα. = I’ll be finding / I’ll have time every day (habitual sense)

No. It is a different stem/form of the same verb.

The dictionary form is:

• βρίσκω = I find

But Greek often uses a different form for perfective meanings:

• να βρω
• θα βρω

So learners need to get used to pairs like this:

• βρίσκω / βρω
• βλέπω / δω
• παίρνω / πάρω

This is very common in Modern Greek and is an important part of how verbs work.

Because Greek does not have a separate indefinite article like English a/an in the same way.

So:

• πτήση can mean flight or a flight, depending on context
• απευθείας πτήση = a direct flight

Greek can use ένας / μία / ένα when it wants to emphasize one or make something more explicitly indefinite, but it is often unnecessary.

So here:

• θα βρω απευθείας πτήση = will I find a direct flight

is perfectly natural.

Here απευθείας means direct or non-stop, and it is being used with πτήση.

So:

• απευθείας πτήση = direct flight

It is a very useful travel word. You may also see απευθείας used adverbially:

• Πήγα απευθείας στο σπίτι. = I went straight/directly home.

So it can function a bit flexibly depending on the sentence.

Not completely. Greek word order is more flexible than English, though some orders sound more natural than others.

This sentence:

• Θέλω να πάω στην Πάτρα το καλοκαίρι, αλλά δεν ξέρω αν θα βρω απευθείας πτήση.

is a very natural neutral order.

You could move some parts for emphasis, for example:

• Το καλοκαίρι θέλω να πάω στην Πάτρα...
• Δεν ξέρω αν απευθείας πτήση θα βρω...
  This is possible, but less neutral and more marked.

So the given version is the most natural everyday phrasing for a learner to imitate.