Σε ποια πόλη θα πάμε το καλοκαίρι;

Σε is the preposition to / in / at. With movement verbs like πάμε (we go), σε often corresponds to English to: to which city.
Grammatically, σε requires the accusative case, so the question word that follows is in accusative (see below).

It’s accusative because it’s the object of the preposition σε.
In feminine singular, ποια (which) looks the same in nominative and accusative, so the form alone doesn’t reveal the case—the preposition does.
You can confirm the noun: πόλη is also identical in nominative and accusative singular, so again you rely on the syntax (σε + accusative).

Because ποια must agree with πόλη in gender, number, and case.

• η πόλη = feminine singular
  So you use ποια (feminine singular).
  (For comparison: ποιος = masculine, ποιο = neuter.)

In Greek, with question words like ποιος/ποια/ποιο, you usually don’t use an article before the noun in this structure.
So Σε ποια πόλη…; is the natural way to say Which city…?
You can get an article in other constructions, e.g. Σε ποια από τις πόλεις…; (Which of the cities…?), but not typically σε ποια την πόλη.

θα is a particle that marks the future. It’s not a verb and it doesn’t change form.
πάμε is the present tense form of the verb πηγαίνω / πάω used with θα to express future:

• θα πάμε = we will go

Both can be correct, but they can suggest different nuance/aspect:

• θα πάμε is the most common for a single, straightforward future trip: we’ll go (a completed move to a destination).
• θα πηγαίνουμε often sounds like we will be going / we will go regularly or emphasizes an ongoing/habitual idea, depending on context.

For a summer trip plan, θα πάμε is the default.

It’s neutral and everyday. It’s not rude.
Ways to sound more formal/planned:

• Σε ποια πόλη θα ταξιδέψουμε το καλοκαίρι; (Which city will we travel to in the summer?)
  Or add a polite framing question:
• Ξέρεις σε ποια πόλη θα πάμε το καλοκαίρι; (Do you know which city we’ll go to in the summer?)

Greek commonly uses the accusative with an article to express time when (like this/that time period):

• το καλοκαίρι = in the summer / during the summer

It’s a standard time expression, not a direct object.

Yes. Word order is flexible and changes emphasis:

• Το καλοκαίρι σε ποια πόλη θα πάμε; (emphasizes in the summer)
• Σε ποια πόλη το καλοκαίρι θα πάμε; (also fine; slightly more emphatic/structured)
  The original Σε ποια πόλη θα πάμε το καλοκαίρι; is very natural.

You don’t need inversion like English (will we). Greek often keeps normal verb order and relies on: 1) a question word (ποια)
2) intonation
3) sometimes punctuation (; in Greek)

So Σε ποια πόλη θα πάμε…; is already clearly a question.

In Greek, ; is the question mark.
A Greek semicolon (roughly like an English semicolon/colon) is · (ano teleia), though modern usage varies.

A practical approximation:

• Σε = seh
• ποια ≈ PYA (often pronounced like one syllable; the ι is very “light”)
• πόλη = POH-lee
• θα = tha (as in this, not thin)
• πάμε = PAH-meh
• το καλοκαίρι = to ka-lo-KAI-ree

Stress is marked by the accent: πόλη, πάμε, καλοκαίρι.

You normally place δεν before θα:

• Σε ποια πόλη δεν θα πάμε το καλοκαίρι; = Which city will we not go to in the summer?
  In more casual speech, δεν may reduce to δε before consonants: δε θα πάμε.