bihaenggireul nohchiji anhge iljjik chulbalhaeyo.

It is built like this:

• 놓치다 = to miss
• -지 않다 = to not do something
• -게 = makes the phrase work like so that, in order to, or in a way that

So 놓치지 않게 means so that (someone) doesn’t miss it or in order not to miss it.

In this sentence, it gives the purpose of leaving early.

Because 비행기 is the direct object of 놓치다.

• 비행기를 놓치다 = to miss the plane/flight

The particle -를 / -을 marks what is being missed.
If you used 비행기가, it would mark 비행기 as the subject, which would not fit here.

Literally, 비행기 means airplane.
But in real Korean, it can also refer to a flight, depending on context.

In 비행기를 놓치다, the natural English meaning is usually to miss the flight, not to miss the physical airplane.

Korean usually puts background information, reasons, and purpose clauses before the main verb.

So the structure is:

• 비행기를 놓치지 않게 = so as not to miss the flight
• 일찍 출발해요 = leave early

Literally, Korean is saying something like:

• So that we don’t miss the flight, we leave early.

That word order is very natural in Korean.

Both can sometimes translate as early/quickly, but they are not the same.

• 일찍 = early, at an early time
• 빨리 = quickly, fast, or soon

Here the idea is about time of departure, not speed, so 일찍 출발해요 means leave early.
If you said 빨리 출발해요, it could sound more like leave quickly or let’s get going soon.

출발해요 comes from:

• 출발하다 = to depart / set off / leave

This is the polite -해요 form.
So 출발해요 means depart, leave, or are leaving, depending on context.

It is a common, neutral, polite form used in everyday conversation.

In Korean, the present polite form often covers:

• present actions
• habitual actions
• near-future plans
• general instructions

So 출발해요 can naturally refer to something you are going to do soon.
Korean does not need a separate future form as often as English does.

The subject is omitted because Korean often drops information that is clear from context.

Depending on the situation, it could mean:

• we leave early
• you leave early
• I leave early

Korean speakers often do not state the subject unless it is needed for clarity or emphasis.

Yes. That is very natural too.

• -게
• -도록

Both can express purpose here: so that / in order not to.

A rough difference is:

• -게 often sounds a little lighter and more conversational
• -도록 can sound a little more explicit or formal

In this sentence, both work well.

Yes, you might hear that in conversation.

• 안 놓치게 = colloquial negative
• 놓치지 않게 = standard negative pattern

Both mean basically the same thing here.
However, 놓치지 않게 often sounds a bit smoother and more standard in careful speech or writing.

By form, 출발해요 is a polite declarative ending, so it is basically a statement.
But in real conversation, Korean often uses this form more flexibly.

Depending on context, it could sound like:

• a plain statement: We leave early so we don’t miss the flight
• a gentle suggestion: Let’s leave early so we don’t miss the flight
• a soft instruction, in some situations

If you want a clearer command, you could say 출발하세요.
If you want a clearer suggestion, you could say 일찍 출발해요 with the right context or use 출발합시다 in more formal speech.