После бассейна детям лучше идти не в сауну, а сразу в душевую.

Because после always takes the genitive case.

• бассейн = nominative
• бассейна = genitive singular

So:

• после бассейна = after the pool / after swimming

This is a fixed grammar rule:

• после урока = after the lesson
• после обеда = after lunch
• после тренировки = after practice

In this sentence, после бассейна means after being in the pool / after swimming.

Детям is dative because the sentence uses an impersonal construction with лучше.

Literally, the structure is something like:

• детям лучше идти... = for the children, it is better to go...

The dative marks the person for whom something is advisable, preferable, necessary, possible, etc.

Compare:

• Мне лучше остаться. = It’s better for me to stay.
• Тебе надо идти. = You need to go.
• Им можно войти. = They may enter.

So детям does not mean the children are doing something to another object; it means it is better for the children to do this.

Лучше идти means it’s better to go or they should rather go.

A few important points:

• лучше literally means better
• With an infinitive, it often expresses advice or recommendation
• It is softer than a direct command

So:

• детям лучше идти не в сауну...
  = children should better go not to the sauna... = more natural English: it’s better for the children to go not to the sauna...

This is a very common Russian way to give advice.

Russian distinguishes between two common verbs of motion:

• идти = to go, to be going, in one direction
• ходить = to go habitually, repeatedly, or in various directions

Here the sentence is about one specific movement right after the pool:

• go to the shower room now, not to the sauna

So идти is the natural choice.

Compare:

• После бассейна детям лучше идти в душевую.
  = After the pool, the children should go to the shower room.
• Дети часто ходят в бассейн по субботам.
  = The children often go to the pool on Saturdays.

Because after в meaning into / to, Russian uses the accusative case to show motion toward a place.

Here the sentence describes movement:

• идти в сауну = to go into/to the sauna
• идти в душевую = to go into/to the shower room

So:

• сауна → в сауну
• душевая → в душевую

Compare with location:

• в сауне = in the sauna
• в душевой = in the shower room

So the rule is:

• motion to a place → в + accusative
• location in a place → в + prepositional

Не ..., а ... is a very common contrast pattern meaning:

• not ..., but ...

So:

• не в сауну, а сразу в душевую
  = not to the sauna, but straight to the shower room

This structure strongly corrects or contrasts two alternatives.

Compare:

• Не чай, а кофе. = Not tea, but coffee.
• Не завтра, а сегодня. = Not tomorrow, but today.

In your sentence, it emphasizes the recommended destination: the shower room, not the sauna.

Сразу means right away, immediately, or straight.

So:

• а сразу в душевую = but straight to the shower room

It adds the idea that there should be no stop in between.

Compare:

• Иди сразу домой. = Go straight home.
• Сразу после урока = Immediately after the lesson.

Here it strengthens the advice: after the pool, the children should go directly to the shower room.

Good question, because both relate to shower, but they are not always identical.

• душ usually means the shower itself or a shower
• душевая usually means the shower room / shower area

So:

• идти в душ can mean go take a shower
• идти в душевую means go to the shower room

In places like swimming pools, gyms, schools, or sports centers, душевая is especially natural because it refers to the room or section.

In school-style grammar, лучше is often treated as a comparative form of хорошо (well) or as a word used in an impersonal predicate.

What matters for learners is how it behaves:

• лучше + infinitive = it is better to...
• кому-то лучше + infinitive = it is better for someone to...

So in this sentence:

• детям лучше идти...

the whole phrase means:

• it is better for the children to go...

You do not need a verb like есть here. Russian often uses this kind of short impersonal predicate without an explicit is.

Yes. Russian word order is flexible, though the original order sounds natural.

Original:

• После бассейна детям лучше идти не в сауну, а сразу в душевую.

You could also say:

• Детям после бассейна лучше идти не в сауну, а сразу в душевую.
• Не в сауну, а сразу в душевую детям лучше идти после бассейна.

But the original is the clearest and most neutral.

Why this order works well:

• После бассейна sets the situation first
• детям introduces who the advice is for
• лучше идти gives the recommendation
• не в сауну, а сразу в душевую gives the contrast

So yes, word order can change, but it changes emphasis.

It sounds most like advice or a recommendation, not a direct command.

• лучше идти = it’s better to go
• A stricter rule might use нужно, надо, or something more direct
• A command would use the imperative

Compare:

• Детям лучше идти в душевую. = It’s better for the children to go to the shower room.
• Детям надо идти в душевую. = The children need to go to the shower room.
• Идите в душевую! = Go to the shower room!

So this sentence sounds like someone giving sensible guidance, perhaps for health or safety reasons.

Yes, пойти is possible, but it changes the nuance slightly.

• идти focuses on the process/direction of going
• пойти focuses more on setting off or starting to go

So:

• лучше идти в душевую = better to go to the shower room
• лучше пойти в душевую = better to head off to the shower room

In many contexts, both are possible. The version with идти sounds very natural in a general recommendation like this one.