Textbooks teach you to order coffee, not to survive a NIE appointment. Here's the language that actually comes up.
Your textbook prepared you to order coffee. It did not prepare you for a NIE appointment.
The NIE and paperwork
Key terms: cita previa (prior appointment), el empadronamiento (registering your address) and el justificante (proof of something). The sentence you will use most is Tengo cita a las diez.
Opening a bank account
Open with Quiero abrir una cuenta bancaria, and be ready for questions about el pasaporte, el NIE and el justificante de domicilio.
Renting a flat
The words that matter: la fianza (deposit), los gastos incluidos (bills included) and el contrato de alquiler (rental contract).
Seeing a doctor
The public health system is la seguridad social; you need a tarjeta sanitaria. Open with Quiero pedir cita con el médico de cabecera.