How to move existing WordPress wp-content folder along with database to new server and new domain name?

Running WP 3.5.1 on a LEMP stack (over at Linode) …

I have a wp-content folder for a WP site along with a full backup of the database. The URL of the site was something like:

test.example.com

I want to get the site operational on my own server at:

test.mydomain.com

And once the site is finished and DNS changes take effect, I want to have the site URL be:

myclientsdomain.com

What is the preferred (and hopefully most straightforward way) of handling server moves and domain changes like this? I’m looking for a step-by-step answer, one that accounts for all the situations described above for both the files and database.

There’s a pretty good step by step on moving WordPress in the Codex. It is what I follow when changing domains.

Moving the files is pretty straight-forward. It is the hard-coded references in the database that are tricky. However, serialized search and replace will take care of all database changes. I’ve used the Velvet Blues plugin in the past, but the Search and Replace script is pretty top-notch.