После прилёта мы не стали брать такси и доехали до центра на автобусе.

Because после requires the genitive case.

• прилёт = nominative
• после прилёта = after the arrival / after arriving

So this is a standard pattern:

• после урока = after the lesson
• после работы = after work
• после прилёта = after the arrival / after arriving

Also note that прилёт is a noun, not a verb. Russian often uses a noun where English might prefer a clause like after we arrived.

Не стали + infinitive is a very common Russian pattern meaning:

• didn’t start to
• didn’t go ahead and
• decided not to

So мы не стали брать такси does not literally focus on the physical act of “starting.” It means something like:

• we decided not to take a taxi
• we didn’t end up taking a taxi

This construction is often softer and more natural in Russian than a very direct negative.

Compare:

• мы не взяли такси = we didn’t take a taxi
• мы не стали брать такси = we decided not to take a taxi / we chose not to take a taxi

The second version suggests a choice.

After не стали, Russian often uses the imperfective infinitive, because the focus is on the action as an option or course of action, not on its completion.

So:

• не стали брать такси = we decided not to take a taxi

This sounds more natural than не стали взять такси, which is not standard here.

A useful way to think about it:

• стать + imperfective infinitive often describes beginning or not beginning an action
• брать is the imperfective partner of взять

So не стали брать is the expected combination.

Because такси is an indeclinable noun in Russian. Its form stays the same in all cases.

So you get:

• такси (nominative)
• брать такси (accusative meaning, but same form)
• на такси
• без такси

Many borrowed words in Russian behave this way. For a learner, the good news is that такси is easy: the case changes in function, but not in spelling.

Доехали is the past plural of доехать. This verb means to get/reach somewhere by transport.

That is a very good fit here, because the sentence is about reaching the city center by bus.

Useful comparison:

• ехать = to be going / to travel by vehicle
• поехать = to set off, to go
• приехать = to arrive
• доехать до... = to get as far as..., to reach ... by transport

So:

• доехали до центра = they reached the center by transport

This emphasizes successful arrival at that point.

Because доехать normally goes with до + genitive.

So the standard pattern is:

• доехать до дома
• доехать до станции
• доехать до центра

This is a set verb-preposition combination.

You certainly can say в центр with other verbs, for example:

• поехали в центр = we went to the center
• приехали в центр = we arrived in the center

But with доехать, Russian usually uses до:

• доехать до центра

Because на автобусе means by bus, while в автобусе means inside the bus.

This is a very important distinction.

• ехать на автобусе = to go by bus
• сидеть в автобусе = to sit in the bus

Russian often uses на + prepositional case for means of transport:

• на автобусе
• на поезде
• на трамвае
• на машине

So here на автобусе tells you the method of transport, not the physical location.

In this sentence, на автобусе is the normal, idiomatic choice.

Russian usually prefers на + vehicle for transport in everyday speech:

• поехать на автобусе
• доехать на автобусе
• ехать на машине

A form like автобусом may appear in some contexts, but it is much less natural here. For a learner, the safest and most standard option is:

• на автобусе

Yes. Russian often omits subject pronouns when the verb ending already makes the subject clear.

So both are possible:

• После прилёта мы не стали брать такси...
• После прилёта не стали брать такси...

Why include мы then? Usually for one of these reasons:

• to make the subject explicit
• to sound a little clearer in a longer story
• to add slight emphasis or contrast

So мы is not required, but it is perfectly natural.

Russian word order is more flexible than English word order, but it is not random.

This sentence starts with После прилёта, which sets the time frame first:

• After arriving...

Then it gives the main subject and actions:

• мы не стали брать такси
• и доехали до центра на автобусе

That order sounds natural in narration: first the situation, then the decision, then the result.

Other word orders are possible, but they may shift emphasis. For example:

• Мы после прилёта не стали брать такси...
  puts мы earlier and sounds slightly more focused on we

• На автобусе мы доехали до центра
  emphasizes the means of transport more

So the original order is natural and neutral.

Yes, you could, but the style is a little different.

• после прилёта = neutral, everyday, very common
• по прилёте = also correct, but often sounds a bit more formal or written

So:

• После прилёта мы не стали брать такси...
  is the most natural all-purpose version

• По прилёте мы не стали брать такси...
  is also correct, but slightly more formal in tone

For most learners, после прилёта is the better default choice.