Otobüs geciktiğine göre, ben yürüyerek gideceğim.

Here -diğine göre means something like since, given that, or judging from the fact that.

So the sentence structure is:

Otobüs geciktiğine göre, ...
= Since the bus is late / delayed, ...
= Given that the bus is late, ...

It is used when the speaker takes one fact and draws a conclusion from it.

It is built from gecikmek = to be late / to be delayed.

A useful breakdown is:

gecik- → verb stem
-dik / -tik → a suffix that turns the verb into a clause-like noun form
-i → 3rd person possessive marker
-n- → buffer consonant
-e → dative case, required by göre

So:

gecik + tik + i + n + e → geciktiğine

You will often see this pattern in Turkish:

verb + -DIK + possessive + case/postposition

In this sentence, the rough literal idea is:

according to its being delayed

Not exactly.

It may look like gecikti = it was late / it got delayed, but in this sentence the form is functioning differently. It is part of a subordinate clause pattern, not just a simple past-tense main verb.

Compare:

Otobüs gecikti.
= The bus was delayed.

Otobüs geciktiğine göre, ...
= Since the bus is delayed, ...

So in the second example, the verb is wrapped into a structure meaning the fact that the bus is delayed and then connected to göre.

Because göre normally takes the dative case.

You can see the same thing in simple noun phrases:

bana göre = according to me
hava durumuna göre = according to the weather

So with a verb clause, Turkish first nominalizes the verb, then adds dative:

geciktiğine göre

That final -e is there because göre requires it.

This is a very common learner question, because in many -DIK clauses the subject does appear in the genitive:

Otobüsün geciktiğini biliyorum.
= I know that the bus is delayed.

But ...diğine göre behaves more like a fixed adverbial pattern, and the subject is very often left in the plain form:

otobüs geciktiğine göre
sen geldiğine göre
hava soğuduğuna göre

So the plain otobüs is natural here. This pattern does not behave exactly like every other -DIK clause.

Yes, it could be left out.

Because gideceğim already means I will go, the subject ben is not grammatically necessary.

So these are both natural:

Otobüs geciktiğine göre, ben yürüyerek gideceğim.
Otobüs geciktiğine göre, yürüyerek gideceğim.

Adding ben gives a little more emphasis or contrast, something like:

As for me, I’ll go on foot.

Yürüyerek means by walking or on foot.

It comes from:

yürümek = to walk
-erek / -arak = a suffix meaning by doing, while doing

So:

yürüyerek = by walking

This suffix is very common:

koşarak geldi = he came running
gülerek konuştu = she spoke smiling/laughing
düşünerek cevap verdi = he answered by thinking carefully

In your sentence, yürüyerek tells you how the speaker will go.

Because the two mean slightly different things.

Yürüyeceğim
= I will walk

This can simply describe the action of walking.

Yürüyerek gideceğim
= I will go by walking / I will go on foot

This emphasizes both:

• the destination idea in gitmek = to go
• the method in yürüyerek = by walking

So if the real point is I will get there on foot, then yürüyerek gideceğim is the more natural choice.

Because the speaker is making a decision about what they will do next.

gideceğim = I will go

That fits the context very well:

• fact: the bus is delayed
• conclusion/decision: I will go on foot

Other tenses would change the feel:

gidiyorum = I’m going
This sounds more immediate or already in progress.

giderim
This can sound more general, habitual, or less definite depending on context.

So gideceğim is the best match for a clear future intention.

Its core meaning is related to according to / in relation to / based on, but the natural English translation depends on context.

For example:

Bana göre doğru.
= According to me, it’s correct.
= In my opinion, it’s correct.

Otobüs geciktiğine göre...
= Since the bus is delayed...
= Given that the bus is delayed...

So in this sentence, göre is not best translated word-for-word as according to. It has more of an inference meaning: the speaker is reasoning from one fact to a conclusion.

Yes. Turkish word order is flexible.

The version you have is very natural because it puts the reason first and the result second:

Otobüs geciktiğine göre, ben yürüyerek gideceğim.

You could also hear:

Otobüs geciktiğine göre yürüyerek gideceğim.
Ben otobüs geciktiğine göre yürüyerek gideceğim.

The main meaning stays the same. What changes is mostly emphasis.

Also, the comma is helpful in writing because the first clause works like an introductory reason clause, but in everyday writing you may sometimes see it omitted.