On the DeliciousBrains.com site they've posted the latest installment for their "Hosting WordPress Yourself" series (part 12) showing how to move your WordPress installation over to a new server.

In part 11 of Hosting WordPress Yourself I demonstrated how to update a server’s packages (including PHP). However, I didn’t show how to upgrade the server itself, because it’s not something I recommend.

[...] In this article, I’m going to walk you through the steps required to migrate an existing WordPress site to a new server. These are the same steps that I took just a few months ago to migrate the Delicious Brains site to a new server with little to no downtime.

The rest of the article is broken up into several different sections, each with descriptions and commands/code you'll need:

• Securely Copying Files
• File Migration
• Nginx Configuration
• HTTPS Certificates
• Spoof DNS
• Database Import

They also include hints on migrating with minimum downtime and things to look out for when you "flip the switch".