taeksireul tago gonghangeuro baro chulbalhalgeyo.

-고 connects actions in sequence: take a taxi and (then) depart.

• 택시를 타고 공항으로 출발할게요 = I’ll take a taxi and head to the airport.
  You could also use -아서/어서 (타서) to mean by doing X, (I) do Y, but 타고 is very natural for and then / by means of.
  If you say only 택시를 타요, it just means I’m taking a taxi (no mention of leaving for the airport).

In this kind of sentence it commonly implies both method and sequence:

• method: go by taxi
• sequence: take a taxi, then leave/head out
  Korean often uses N을/를 타고 to express transportation: 버스를 타고, 지하철을 타고, 택시를 타고.

-으로/로 marks direction: toward/to a place.
-에 often marks a destination/arrival point or location.
With verbs like 가다/오다/향하다/출발하다, -으로 is very common because it emphasizes heading in that direction:

• 공항으로 출발하다 = depart toward the airport
  You can sometimes hear 공항에 가다 (go to the airport), but with 출발하다, 공항으로 sounds more natural.

Not exactly:

• 출발하다 = to depart / set off / leave (focus on the moment of leaving)
• 가다 = to go (focus on the movement/going)
  So 공항으로 출발할게요 emphasizes I’m leaving now (headed for the airport).

-(으)ㄹ게요 expresses the speaker’s intention/decision, often made now or presented as a personal commitment. It can sound like:

• I’ll leave (then).
  It’s common in conversation when telling someone what you’ll do next, especially if it affects them or is a response to the situation.

Both can translate as I will depart/leave, but the feel is different:

• 출발할게요: more immediate intention/decision, often like Okay, I’ll leave now/then.
• 출발할 거예요: more neutral future statement/plan, like I’m going to leave (later / as a plan).
  In many real conversations, 할게요 sounds more like “I’ll do it (now/for you/then).”

Korean often omits the subject when it’s obvious. 출발할게요 already strongly implies I (the speaker).
You can add it for emphasis or clarity:

• 저는 택시를 타고 공항으로 바로 출발할게요.
  But it’s not required.

It’s polite informal (often called 해요체). It’s appropriate for most everyday situations (store staff, coworkers, people you don’t know well).
More formal alternatives:

• 출발하겠습니다 (more formal/announcing)
• 출발합니다 (plain formal statement)

Yes, 바로 is flexible, but placement changes what it emphasizes:

• 공항으로 바로 출발할게요: depart immediately (toward the airport)
• 바로 택시를 타고 공항으로 출발할게요: immediately take a taxi and depart
  Your original version is very natural.

In fast speech, it often sounds like:

• 출발할게요 → roughly 출발할께요 (many speakers tense the ㄱ sound)
  And 할게요 is commonly heard as a smooth chunk 할게요.
  (Spelling stays 출발할게요 in standard writing.)