Я пойду спать пораньше, чтобы утром чувствовать себя бодрее.

Yes and no.

• Пойду is the 1st person singular future of the perfective verb пойти.
• Идти is imperfective, and пойти is its perfective partner in many contexts.
• Because пойти is perfective, it forms the future with a simple future:
  я пойду = I will go

In this sentence, though, пойду is part of the common pattern пойти + infinitive, which often means:

• to go and do something
• to set off to do something
• or very naturally in English, to go to ...

So я пойду спать means I’ll go to sleep / I’ll go to bed, not just I will walk.

Because these mean different things.

• Я пойду спать = I’ll go to bed / I’ll go to sleep
• Я буду спать = I’ll be sleeping / I will sleep

So:

• пойду спать focuses on the decision or action of going to bed
• буду спать focuses on the state of sleeping

In your sentence, the speaker means they will go to bed earlier, so пойду спать is the natural choice.

Пораньше means a bit earlier, earlier than usual, or somewhat early.

It comes from раньше = earlier.

The prefix по- often softens the meaning and gives the sense of:

• a little
• somewhat
• rather
• earlier than normal / than planned

So:

• раньше = earlier
• пораньше = a little earlier / nice and early / earlier than usual

In this sentence, пойду спать пораньше sounds very natural:
I’ll go to bed a bit earlier / earlier than usual.

Here чтобы means so that or in order to.

It introduces the purpose of the first action:

• Я пойду спать пораньше = I’ll go to bed earlier
• чтобы утром чувствовать себя бодрее = so that I feel more energetic in the morning

So the whole sentence expresses:

• first, the action
• then, the reason or goal for doing it

This is one of the most common uses of чтобы in Russian.

Because in Russian, after чтобы, if the subject is the same as in the main clause, an infinitive is very common.

Here the subject is я in both parts:

• Я пойду спать пораньше
• so that I feel more energetic in the morning

Since the subject stays the same, Russian naturally uses the infinitive:

• чтобы утром чувствовать себя бодрее

You could also say something like:

• чтобы утром я чувствовал себя бодрее

But that sounds more explicit and a bit heavier. The infinitive version is very natural and elegant here.

Чувствовать себя is the standard Russian way to say to feel in the sense of to feel oneself / to feel in a certain condition.

So:

• чувствовать by itself = to feel, to sense
• чувствовать себя = to feel as in to feel well, tired, energetic, sick, etc.

Examples:

• Я чувствую запах. = I smell / sense a smell.
• Я чувствую себя хорошо. = I feel good.

In your sentence:

• чувствовать себя бодрее = to feel more energetic / more alert

Grammatically, себя is the reflexive pronoun oneself.

Here себя is in the accusative case.

That is because чувствовать normally takes a direct object:

• чувствовать что? = to feel what?

In чувствовать себя, the object is oneself, so Russian uses the reflexive pronoun себя.

This form себя can look the same in different cases, but in this expression it is functioning as the direct object, so it is best understood here as accusative.

Бодрее means more energetic, more alert, more refreshed, or less tired.

It is the comparative form of бодрый:

• бодрый = energetic, lively, alert
• бодрее = more energetic / more alert

Why comparative? Because the idea is:

• if I go to bed earlier,
• then in the morning I will feel more energetic than otherwise

Russian often uses the comparative this way when speaking about improvement:

• лучше = better
• хуже = worse
• бодрее = more energetic

So чувствовать себя бодрее is a very natural phrase.

Because Russian often uses certain time expressions in a bare case form, without a preposition.

Утром means in the morning.

This is an adverbial use of the instrumental form:

• утро = morning
• утром = in the morning

Similarly:

• днём = in the daytime
• вечером = in the evening
• ночью = at night

So утром is just the normal way to say in the morning here.

Yes. Russian word order is flexible, though different orders can shift emphasis slightly.

The original sentence:

• Я пойду спать пораньше, чтобы утром чувствовать себя бодрее.

A very natural alternative is:

• Чтобы утром чувствовать себя бодрее, я пойду спать пораньше.

This puts more emphasis on the purpose first.

You could also move пораньше for nuance, but the original order sounds very natural and neutral.

So the sentence order is not fixed in the same way as English, but the version you have is completely standard.

Literally, спать means to sleep.

But in the common expression идти / пойти спать, the whole phrase usually means:

• to go to bed
• to go to sleep

So even though the second verb is literally sleep, the full expression is idiomatic.

That is why:

• Я иду спать. = I’m going to bed.
• Я пойду спать. = I’ll go to bed.

English and Russian package this idea a little differently, but this is one of the most common everyday Russian expressions.