Learn Hardcore Japanese: zyugyouryou no siharai no tame ni, ginkou ni ikimasu.

Each の links two nouns into a larger noun phrase. The first の connects 授業料 (tuition) and 支払い (payment) to form “payment of tuition.” The second の links that entire noun phrase (授業料の支払い) with ため (sake, purpose), yielding “for the purpose of paying tuition.” You need both to express those two relationships.

What does ために mean here, and why is it used instead of just ため?

ため by itself is a noun meaning “sake” or “benefit,” but when you want to express purpose (“in order to…”) you add に to make it an adverbial phrase. So ～のために means “for the sake of…,” “in order to…,” and it indicates the goal or purpose of the action that follows.

Could I rephrase this sentence using the verb 支払う instead of the noun 支払い?

Yes. You can say:
授業料を支払うために、銀行に行きます。
Here 支払う (to pay) is in its plain dictionary form plus ために, directly expressing the purpose. It’s more common in speech, whereas ～の支払いのために is a bit more formal or written style.

What’s the difference between 授業料を支払うために and 授業料の支払いのために?

Both mean “in order to pay tuition,” but:

授業料を支払うために uses a full verb clause (～を支払う) before ために. It feels direct and conversational.
授業料の支払いのために turns 支払う into the noun 支払い (a nominalization), then attaches の
- ために. It’s more nominal/abstract and often seen in written or formal contexts.

Why is the particle に used after 銀行 (銀行に行きます) rather than で or へ?

The particle に marks the destination of movement: “go to the bank.”

に = destination/location (I go to the bank).
へ also indicates direction (“toward the bank”) but with a slightly more abstract nuance.
で marks where an action takes place (“pay at the bank”) rather than where you go.

Can I shorten 授業料の支払いのために to 授業料支払いのために?

Yes, in many contexts you can drop the first の, yielding 授業料支払いのために. It’s a more compact, written-style form. In spoken or very clear writing, including both の particles can make the relationships clearer.

Why is there a comma before 銀行に行きます?

Japanese often uses commas (読点) to separate long or complex phrases for readability, much like in English. Here, the comma marks the end of the purpose clause (授業料の支払いのために) and the start of the main clause (銀行に行きます).

Could I use ので or から instead of ために to express “because” or “in order to”?

ので and から both express reason/cause (“because”), not purpose.
ために expresses purpose (“in order to…”).
If you said 授業料の支払いのために…, you imply you’re doing the next action so that the payment can happen. If you said 授業料の支払いから or ので, it would sound like you are giving the reason why you went to the bank (still “because of the payment”), but it doesn’t carry the nuance “in order to pay.” For a clear purposive nuance, stick with ために.