Forgive me if this is a silly question, but from a developer perspective, are Invoices and Sales Orders simply data values? And do we not need to interact with a bank API and things like that? Is that something that NetSuite does internally when changing the stage type?

They are both transactions. You don't have to have bank integration, but you can if you wish. Payments are obviously different record types, as are payment files, cash payments etc. It's a gigantic system & impossible to answer your very broad question in a little slack reply. What are you actually wanting to know/learn? It sounds like you a very new to NetSuite

This makes sense, I'm mostly wanting to learn what happens in the background when the records are created, be it a Sales Order, Invoice, Payment etc.

If you give some indication about what you actually want to do/are trying to do, we can steer you in the right direction

NetSuite is mostly an accounting, CRM, or inventory management system to put it too simply. When transaction records are created there are some native things going on in the background like status links between records, item lines and GL Impact for example. Also this is oversimplified. It's too broad to understand in a nutshell without experience. I suggest you watch the broad introduction videos about ERP to begin with, then follow that with some introduction to SuiteScript training and then try to make a few customisations across the Sales Order, Fulfillment and Invoice transactions. Bonus points for looking at the GL Impact subtab and the Accounts register.