Чтобы быстрее добраться до вокзала, нам нужно пройти через площадь.

Here чтобы means in order to or so as to. It introduces the purpose of the action:

Чтобы быстрее добраться до вокзала...
= To get to the station faster...

So the first part of the sentence explains why they need to go through the square.

Быстрее is the comparative form of быстро (quickly).

• быстро = quickly
• быстрее = faster / more quickly

Russian often prefers this simple comparative form instead of something like более быстро.

So быстрее добраться means to get there faster.

Добраться до means to reach, to get to, or to make one’s way to a place, often with some sense of effort or travel.

So:

• добраться до вокзала = to get to the station

This verb normally goes with до + genitive for the destination.

Also, добраться is perfective, so it focuses on the result: actually reaching the station.

Because the verb добраться requires the preposition до plus the genitive case.

• вокзал = nominative
• вокзала = genitive

So:

• добраться до вокзала = to get to the station

You should learn добраться together with its pattern:

добраться до + genitive

Because нужно is used with the dative case to mean it is necessary for someone.

• нам = to us / for us
• нужно = necessary / need to

So нам нужно пройти literally means something like for us it is necessary to go through, which in natural English is we need to go through.

This is a very common Russian pattern:

• мне нужно = I need
• тебе нужно = you need
• нам нужно = we need

Пройти is perfective and fits the idea of completing a route or passing through a place.

• идти = to be going / to go on foot
• пройти = to go through, pass through, walk across, complete that stretch of movement

Here the meaning is not just to be walking, but to go through the square as part of reaching the station. That is why пройти works well.

So нам нужно пройти через площадь means we need to go across / through the square.

Через means through or across.

It is used when movement crosses an area, space, or obstacle.

So:

• через площадь = across the square
• через мост = across the bridge
• через парк = through the park

In this sentence, the idea is that they must cross the square on the way to the station.

Good question. Через does take the accusative case, but площадь is one of those nouns whose nominative and accusative singular forms look the same.

So here:

• nominative: площадь
• accusative: площадь

Even though the form does not change, the case is still accusative because that is what через requires.

Because Чтобы быстрее добраться до вокзала is a subordinate clause of purpose, and Russian separates such clauses with a comma.

So the structure is:

• subordinate clause: Чтобы быстрее добраться до вокзала
• main clause: нам нужно пройти через площадь

Russian punctuation is often stricter than English in this kind of sentence.

Yes. Russian word order is flexible.

For example, you could also say:

Нам нужно пройти через площадь, чтобы быстрее добраться до вокзала.

This has essentially the same meaning: We need to go through the square to get to the station faster.

The original version puts the purpose first, which gives a little more emphasis to getting to the station faster.

Because each infinitive has a different job in the sentence.

• добраться belongs to the purpose clause: to get to the station faster
• пройти depends on нужно: we need to go through the square

So the sentence is built like this:

• To get to the station faster
• we need to go through the square

Using infinitives in both parts is very natural in Russian when the subject is understood from context.

In this sentence, English would most naturally say across the square, because a square is usually an open space.

Russian uses через for this kind of crossing movement, and depending on the noun, English may translate it as either through or across.

So:

• через площадь = usually across the square
• через лес = through the forest
• через реку = across the river