بعد الرحلة أعود إلى البيت.

بعد means after.

So بعد الرحلة means after the trip or after the journey.

A useful grammar point: بعد is often followed by a noun, and together they form a structure similar to after + noun in English. In fully vowelled MSA, this would be:

بَعْدَ الرِّحْلَةِ

Here, الرحلة is in the genitive because it comes after بعد in this construction.

Because the verb itself already tells you the subject.

أعود means I return / I go back.
The initial أـ on the verb marks first person singular in the present tense.

So Arabic does not need أنا here. You could say أنا أعود إلى البيت, but it is usually unnecessary unless you want emphasis.

أعود is in the present tense in Arabic, but the Arabic present tense often covers several meanings that English separates:

• I return
• I am returning
• I do return
• sometimes I will return, depending on context

In this sentence, because of after the trip, English may naturally translate it as a future idea, but the Arabic form is still the regular present.

If you want to make the future explicit, you could say:

• سأعود إلى البيت = I will return home
• سوف أعود إلى البيت = I will return home

إلى means to.

The verb عاد / يعود often takes إلى when you say where someone returns to:

• أعود إلى البيت = I return to the house/home
• عاد إلى المدينة = He returned to the city

So إلى is the normal preposition here.

It can mean house, but in many sentences like this it is naturally understood as home.

So:

• أعود إلى البيت can be understood as I return to the house
• but very often it means I return home

This is one of those places where Arabic uses a noun with the, while English may prefer the idiomatic expression home without an article.

Arabic usually expresses this idea with a noun, often البيت.

English says I go home without to the, but Arabic commonly says:

• أذهب إلى البيت
• أعود إلى البيت

So even when English uses home by itself, Arabic often uses to the house/home structurally.

Because the sentence is talking about a specific, understood place: the house / the home.

In Arabic, that is very natural. Even when English would simply say home, Arabic often uses the definite form:

• البيت = the house / the home

If you wanted to be more explicit and say my home, you could say:

• بيتي = my house / my home

So أعود إلى البيت is natural, and أعود إلى بيتي is also possible if you want to emphasize my.

Yes. Arabic word order is flexible.

The given sentence:

بعد الرحلة أعود إلى البيت

starts with the time phrase after the trip, which gives that information first.

You could also say:

أعود إلى البيت بعد الرحلة

Both are correct. The difference is mostly one of emphasis and style:

• بعد الرحلة أعود إلى البيت = emphasizes after the trip
• أعود إلى البيت بعد الرحلة = starts with the action I return

A careful pronunciation with vowels would be:

baʿda r-riḥlati aʿūdu ilā l-bayti

A few helpful notes:

• بعد = baʿda
• الرحلة is pronounced ar-riḥla or r-riḥlati in connected speech, because ر is a sun letter, so the l of الـ is not pronounced
• أعود begins with ʿ, the Arabic letter ع, which has no exact English equivalent
• البيت keeps the l sound in الـ, because ب is a moon letter

In fully vowelled MSA, the sentence is:

بَعْدَ الرِّحْلَةِ أَعُودُ إِلَى الْبَيْتِ

The endings are:

• بعدَ: accusative as a time expression
• الرِّحلةِ: genitive
• أعودُ: nominative, because it is a present-tense verb not made subjunctive or jussive by anything
• البيتِ: genitive after إلى

If you are still learning basic reading, you do not need to produce all of these endings all the time, but it is good to know what they are.

Yes. A common alternative is أرجع.

So you could also say:

بعد الرحلة أرجع إلى البيت

That also means After the trip, I return/go back home.

In MSA, أعود is very standard and common. أرجع is also common and understandable. A learner should recognize both.