Changing domain name

I’m going to changing the domain name for my WP site, I’m not copying it over to a different server or anything, but the domain name is going to change. I have read the instructions on how to do this, and read about the Search and Replace for WordPress Databases Script, but I have some (very basic) questions.

1. So, basically the 1st step is to back up the database.
2. After that I need to run the script; then
3. Move it to the new domain

I’m just confused about the order. Do I need to backup the database, then move to the new domain, and THEN run the script or do I run the script before I before I move to the new domain?

I know this might seem like a ridiculously stupid question, but I can’t wrap my head around this. Please help, ’cause I’m going to freak out.