В субботу мы собираемся в поход, и моя сестра хочет разжечь костёр у реки.

Because after в when you mean on Saturday / at that time, Russian uses the accusative for days of the week.

The base form is суббота.
The accusative singular is субботу.

So:

в субботу = on Saturday

This is a very common pattern:

в понедельник = on Monday
в среду = on Wednesday
в пятницу = on Friday

So here в субботу answers the question when?

Here, собираемся means something like we are getting ready, we are planning, or we intend.

The verb собираться can have more than one meaning depending on context:

собираться = to gather / get together
собираться + infinitive = to be going to do something
собираться в поход / в дорогу = to get ready to go on a hike / trip

So in this sentence, мы собираемся в поход means we’re getting ready to go on a hike or we’re planning a hike, not literally we are gathering.

Context is what makes the meaning clear.

The -ся ending shows that the verb is reflexive.

The non-reflexive verb is собирать = to collect / gather something.
For example:

собирать вещи = to pack / gather things

But собираться often means:

to get ready
to plan
to intend

So:

мы собираемся = we are getting ready / we are planning

This is one of those cases where the reflexive form has developed its own common meaning and should be learned as a separate verb.

Because Russian idiomatically says идти в поход, собираться в поход, поехать в поход.

Even though English says go on a hike, Russian uses в here, not на.

So you should learn в поход as a set expression:

идти в поход = to go on a hike / camping trip
собираться в поход = to get ready for a hike / plan a hike

This is a good reminder that Russian prepositions do not always match English ones.

Because в takes the accusative when it shows direction or movement toward a goal.

Here the idea is not static location, but entering or going into an activity/trip:

в поход

The noun поход is masculine and inanimate, and its accusative singular looks the same as the nominative singular:

поход → поход

So the form does not change visibly, but the case is still accusative.

Compare:

в лес = into the forest
в город = into the city
в поход = on/into a hike, on a hiking trip

This is about aspect.

разжечь is perfective. It focuses on the result: to light / get the fire started.

разжигать is imperfective. It focuses more on the process, repetition, or ongoing action: to be lighting, to light repeatedly, to light in general.

After хотеть, either aspect can be possible, depending on meaning. Here the sister wants to achieve a specific result:

хочет разжечь костёр = she wants to light the campfire

That is why the perfective разжечь is very natural here.

Костёр means campfire or bonfire.

It is not just any fire. For example:

огонь = fire in a general sense
пожар = a fire as in a blaze/fire emergency
костёр = a campfire / bonfire

So разжечь костёр is a very common phrase meaning to light a campfire.

Also, note the stress: костёр.
The ё is important here and is always stressed.

Because the preposition у means by, near, or next to, and it requires the genitive case.

The dictionary form is река.
Its genitive singular is реки.

So:

у реки = by the river / near the river

This is a very common pattern:

у дома = by the house
у школы = by the school
у моря = by the sea

Because к реке means toward the river / to the river, while у реки means by the river.

So the difference is:

к реке = movement toward the river
у реки = location near the river

In this sentence, the campfire is located by the river, so у реки is the correct choice.

Because this sentence contains two separate clauses:

В субботу мы собираемся в поход
моя сестра хочет разжечь костёр у реки

Each clause has its own subject and verb:

мы собираемся
моя сестра хочет

When и joins two independent clauses like that, Russian normally uses a comma.

So the comma is correct and expected here.

Yes. Russian word order is fairly flexible, although different orders sound more natural in different contexts.

The sentence as given is neutral and natural:

В субботу мы собираемся в поход, и моя сестра хочет разжечь костёр у реки.

You could move things around for emphasis, for example:

Мы в субботу собираемся в поход...
This puts a little more attention on we.

У реки моя сестра хочет разжечь костёр.
This emphasizes the location more strongly.

But the original order is a very normal default order for everyday Russian.

It usually does not show them directly.

Russian has no articles like a and the. So:

в поход can mean on a hike or on the hike, depending on context
костёр can mean a campfire or the campfire
у реки can mean by a river or by the river

The listener understands the intended meaning from context, not from articles.

That is why the Russian sentence sounds complete even though there is no separate word for a or the.