Vespere ni iras malrapide laŭ la rivero por trankviliĝi.

Because vespere is an adverb. In Esperanto, -e marks adverbs, and time expressions are often made this way.

• vespero = evening
• vespere = in the evening, in the evenings, evening-time

So Vespere ni iras... means something like In the evening, we go... or Evenings, we go...

Yes, en la vespero is possible, but vespere is usually more natural and compact.

Esperanto often prefers simple adverbs for time:

• matene = in the morning
• tage = by day, in the daytime
• nokte = at night
• vespere = in the evening

So vespere is a very normal choice here.

It is there for emphasis and framing. Putting vespere first sets the time right away: As for the evening...

Esperanto word order is fairly flexible, so you could also say:

• Ni iras vespere malrapide laŭ la rivero por trankviliĝi.

That would still be correct, but the original version highlights the time first.

Because Esperanto normally uses the simple present tense -as where English might use either go or are going.

So ni iras can mean:

• we go
• we are going
• sometimes even we’re going in a planned sense, depending on context

The longer form estas irantaj is grammatical, but it is much less common and usually used only when the ongoing nature really needs emphasis.

Because it describes how the action of going happens. It modifies the verb iras, so it must be an adverb.

• rapida = fast
• malrapida = slow
• malrapide = slowly

If you used an adjective, it would need to describe a noun instead.

It has three parts:

• rapid- = fast
• mal- = opposite
• -e = adverb ending

So:

• rapide = quickly
• malrapide = slowly

The prefix mal- is very common in Esperanto for forming opposites.

Laŭ means along, following the line of, or following the course of something.

So laŭ la rivero means along the river.

It suggests movement that follows the river’s edge or direction, not just being near it.

This is a useful distinction:

• laŭ la rivero = along the river
• apud la rivero = beside the river, next to the river

So laŭ focuses on the path of movement, while apud focuses on location near something.

Because after a preposition like laŭ, the noun is normally in its basic form, without -n.

So:

• laŭ la rivero = along the river

The accusative -n is not used here because laŭ already gives the needed relationship. In Esperanto, directional -n is mainly used with prepositions that otherwise express location, such as:

• en la domo = in the house
• en la domon = into the house

But laŭ already expresses movement along a path, so laŭ la rivero is the normal form.

Because the sentence refers to a specific river, probably one known from the situation or context.

• la rivero = the river
• rivero by itself would mean just a river in a very general sense, but Esperanto usually does not use a separate indefinite article like English a

So la rivero feels natural if the speakers have a particular river in mind.

Because por expresses purpose: for, in order to, so as to.

So:

• por trankviliĝi = in order to calm down

This tells you why they are walking along the river.

It means to become calm, to calm down, or to grow tranquil.

It is built like this:

• trankvila = calm, tranquil
• -iĝ- = become, get into a state
• -i = infinitive

So trankviliĝi literally means to become calm.

Because the two forms mean different things:

• trankviliĝi = to become calm, to calm down
• trankviligi = to calm someone or something

The suffixes matter:

• -iĝ- = become
• -ig- = cause to become

Here the meaning is that we go along the river so that we ourselves become calm, so trankviliĝi is the right choice.

Yes, in meaning, but not by using a separate reflexive pronoun.

In Esperanto, -iĝ- often carries the idea of a change happening to the subject itself. So trankviliĝi already gives the sense of calm oneself down / become calm.

If you wanted to say to calm us, you could instead say:

• por trankviligi nin

That would be more explicitly causative.

Yes. Esperanto allows fairly flexible word order, especially when the grammar is clear.

For example, you could also say:

• Ni iras malrapide laŭ la rivero vespere por trankviliĝi.
• Ni vespere iras laŭ la rivero malrapide por trankviliĝi.

The original sentence sounds natural because it starts with the time expression and then moves through the action, manner, path, and purpose in a clear way.