Having now moved several WordPress sites from their test environments to new domains, I found the official instructions for moving a WordPress site to be confusing. Here are the steps I follow:

1. Export the old database to SQL file.
2. Run the SQL file through this online search and replace utility. This will update all the links, including the media files, to the new domain.
3. Create a new empty database for the new domain.
4. Move the entire WordPress directory to the new host.
5. Update the wp-config.php file with the information needed to access the new database.
6. Update the permalinks to the default and press save. Then return the permalinks to the way you want them.

That should do it.