The offer is signed, the flat is half-found, and HR has sent a short list: NIE, social security number, IBAN, address. Each item looks like a single errand. Together they form a chain, and the chain only runs in one direction.
Start with the number. If you are an EU citizen moving for good, the document to aim for is the EU registration certificate: it registers you as a resident and carries your NIE. Book the appointment for the certificate rather than for the NIE assignment, and bring your contract as proof of your ground. If you do not live here yet, the EX-15 gives you the number alone, at a police desk in Madrid or at a Spanish consulate before you travel.
For non-EU hires the order shifts. The right to work comes from your authorisation, which the employer usually helps to prepare through the Oficina de Extranjería on Calle Silva; the TIE then follows the toma de huellas in Madrid. Be wary of anyone who says a NIE alone lets you start.
Next comes the social security number. Some HR teams handle it while onboarding you; some leave it to you. Either way, keep your NIE document at hand, because the request starts from it.
The bank account and the padrón close the loop. Banks open an account in your name more readily once you have the NIE, and the empadronamiento at the ayuntamiento gives you the certificate that schools, health centres and later procedures ask for. If you rent, check before signing that your contract, or an authorisation from the landlord, will be accepted for the padrón.
The one thing to avoid is running the steps in parallel and hoping. A bank appointment without a NIE is wasted, and so is a padrón appointment with a lease that is not yet in your name. Line them up one after another, and each step clears the way for the next.