Я откладываю деньги на отпуск.

Yes, я can be omitted if the subject is clear from context.

Russian verbs already show the person, and откладываю means I set aside / I am setting aside, so Откладываю деньги на отпуск is often perfectly natural in conversation.

Including я can:

• make the subject explicit,
• add contrast, as in Я откладываю деньги, а он всё тратит,
• or simply sound a bit more complete out of context.

So both are possible:

• Я откладываю деньги на отпуск.
• Откладываю деньги на отпуск.

Here откладываю means I am setting aside or saving up money little by little for a purpose.

This verb can confuse learners because откладывать also has another common meaning: to postpone.

For example:

• откладывать встречу = to postpone a meeting
• откладывать деньги = to set aside money

So the object often tells you which meaning is intended. With деньги, the meaning is usually to save / put aside.

In Russian, the present tense of an imperfective verb is often used for:

• actions happening now,
• repeated actions,
• habits,
• ongoing processes.

So Я откладываю деньги на отпуск does not have to mean I am literally doing it at this second. It can also mean:

• I’m saving money for a vacation
• I’ve been setting money aside for a vacation
• I do this regularly

This is a very normal use of the imperfective present.

Yes, откладываю is the imperfective form.

The usual perfective partner is отложу / отложить.

Compare:

• Я откладываю деньги на отпуск. = I am saving / I save money for a vacation.
  
  • ongoing, repeated, process-focused
• Я отложу деньги на отпуск. = I will set aside money for a vacation.
  
  • one future completed action, or a result-focused action

Very often, when talking about a continuing savings process, Russian prefers the imperfective:

• Я откладываю деньги...

Деньги is a very important word to notice because it is normally used only in the plural in modern Russian.

So Russian says:

• деньги = money

Even though English money is grammatically singular/uncountable, Russian treats деньги as a plural noun.

That is why you get:

• деньги = nominative plural
• Я откладываю деньги = accusative plural

For inanimate plural nouns like деньги, nominative and accusative are the same in form.

Деньги is in the accusative case because it is the direct object of the verb откладываю.

You are doing the action to the money:

• откладываю что? → деньги

Because деньги is an inanimate plural noun, its accusative form looks the same as the nominative:

• nominative: деньги
• accusative: деньги

So the case is accusative, even though the form does not change.

In this sentence, на + accusative expresses purpose very naturally: money set aside for something.

So:

• деньги на отпуск = money for a vacation
• деньги на билет = money for a ticket
• деньги на еду = money for food

You can sometimes use для in Russian, but it often sounds more like for the sake of, intended for, or just less idiomatic in this kind of everyday savings context.

So:

• откладываю деньги на отпуск = the most natural phrasing here

It is in the accusative singular.

The preposition на can take different cases depending on meaning, but here it means something like for / toward the purpose of, so it takes the accusative.

The noun отпуск is masculine and inanimate:

• nominative: отпуск
• accusative: отпуск

Again, because it is inanimate, the accusative looks the same as the nominative.

So even though the form does not change, the case here is accusative.

Yes, absolutely. Коплю деньги на отпуск is very natural and common.

There is a small nuance:

• копить деньги = to save up / accumulate money
• откладывать деньги = to put money aside regularly

In many contexts they are very close and both work well.

Roughly:

• Я коплю деньги на отпуск emphasizes building up savings.
• Я откладываю деньги на отпуск emphasizes setting some money aside from what you have.

In everyday speech, both are common and often interchangeable.

Yes. Russian word order is flexible, though the neutral order here is very natural:

• Я откладываю деньги на отпуск.

Other orders are possible, depending on emphasis:

• На отпуск я откладываю деньги.
  Emphasizes for a vacation
• Деньги я откладываю на отпуск.
  Contrasts money with something else
• Откладываю деньги на отпуск.
  Natural when I is understood

So the basic meaning stays the same, but the focus or tone can shift.

The main word stress is:

• Я откла́дываю де́ньги на о́тпуск.

A rough pronunciation guide:

• я = ya
• откла́дываю = at-KLA-di-va-yu
• де́ньги = DYEN-gi
• на = na
• о́тпуск = OT-poosk

A learner should especially notice:

• де́ньги has stress on the first syllable.
• о́тпуск also has stress on the first syllable.
• откла́дываю has the stress on кла́.

Not really in normal usage.

Even though откладывать can mean to postpone, the phrase откладывать деньги is a fixed and very common expression meaning to set aside money.

A native speaker will understand:

• откладывать встречу = postpone a meeting
• откладывать поездку = postpone a trip
• откладывать деньги = save/set aside money

So in this sentence, the meaning is clear from the object деньги.