После прогулки мы вернёмся домой на трамвае.

Because после requires the genitive case.

The dictionary form is прогулка = walk / stroll.
Its genitive singular form is прогулки.

So:

• прогулка = a walk
• после прогулки = after a walk / after the walk

This is a very common pattern:

• после работы = after work
• после урока = after the lesson
• после обеда = after lunch

It can mean either. Russian has no articles, so прогулки by itself does not tell you whether English should use a or the.

You decide from context:

• После прогулки мы вернёмся домой could be
  After a walk, we’ll return home
  or
  After the walk, we’ll return home

Russian leaves that distinction unstated unless the context makes it clear.

Вернёмся means we will return / we will come back.

It comes from the verb вернуться = to return, to come back.

This form is:

• 1st person plural = we
• future tense
• from a perfective verb

So мы вернёмся = we will return.

The -ся is part of the verb вернуться.

Compare:

• вернуть = to return something, to give something back
• вернуться = to return oneself, to come back

So:

• Я вернул книгу. = I returned the book.
• Я вернулся домой. = I came back home.

In this sentence, we need the intransitive idea come back, so вернуться / вернёмся is correct.

This is because вернуться is a perfective verb.

In Russian, perfective verbs do not have a true present tense. Their present-looking forms actually refer to the future.

So:

• вернуться → вернусь, вернёшься, вернёмся
  = I will return, you will return, we will return

If you wanted an imperfective future such as will be returning or a repeated/habitual sense, you would use a different structure, for example:

• будем возвращаться = we will be returning / we will return repeatedly

But here the sentence describes one completed future action, so вернёмся is the natural choice.

Yes. In many Russian texts, ё is often written as е, so you may also see вернемся.

But the pronunciation is still ver-NYOM-sya, not ver-NEM-sya.

So:

• вернёмся = fully marked spelling
• вернемся = common unmarked spelling

For learners, it is helpful to remember the ё, because it shows both the pronunciation and the stress.

Домой is a special adverb meaning homeward / to home / going home.

Russian distinguishes:

• дома = at home
• домой = to home / homeward

So:

• Мы дома. = We are at home.
• Мы идём домой. = We are going home.

In this sentence, there is movement toward home, so домой is exactly the right word.

В дом usually means into the house/building, focusing on entering a physical structure, not simply going home in the everyday sense.

Here на трамвае means by tram / on the tram.

With many forms of transport, Russian commonly uses на + prepositional case:

• на трамвае = by tram
• на автобусе = by bus
• на поезде = by train
• на машине = by car

So трамвае is the prepositional singular form of трамвай.

Even though English often says by tram, Russian uses a structure that literally feels more like on the tram.

Usually, in everyday Russian, на трамвае is the normal and most natural choice.

A bare instrumental such as трамваем is much less common here and can sound bookish, formal, or unnatural depending on context.

For a learner, the safest pattern is:

• ехать на трамвае
• вернуться на трамвае
• добраться на трамвае

So in this sentence, на трамвае is the form you should expect and use.

The word order can change. Russian word order is fairly flexible, and changes often affect emphasis rather than basic meaning.

The original sentence:

• После прогулки мы вернёмся домой на трамвае.

This is a natural, neutral sentence. It starts with the time phrase после прогулки = after the walk, setting the scene first.

Other possible orders include:

• Мы вернёмся домой на трамвае после прогулки.
• Домой мы вернёмся на трамвае после прогулки.

These are grammatically possible, but they shift the focus a bit. The original version sounds very normal and straightforward.

Strictly speaking, мы is not always necessary, because вернёмся already shows 1st person plural.

So both are possible:

• Мы вернёмся домой на трамвае.
• Вернёмся домой на трамвае.

However, Russian often keeps the pronoun when it wants to be clear, neutral, or slightly emphatic. In a full sentence like this, мы sounds completely natural.

Both could be possible in some contexts, but they are not identical.

• вернёмся = we will return / come back
• поедем = we will go / set off by vehicle

Вернёмся emphasizes the idea of coming back after being somewhere else.
That fits well with после прогулки: after the walk, we’ll return home.

If you said:

• После прогулки мы поедем домой на трамвае.

that would mean After the walk, we’ll go home by tram, which is also possible, but it focuses more on the trip itself than on the idea of returning.