It's important to treat content management systems like any other application and segment development, test, and production. Before adding content and attaching files to nodes, it's very important to set the file system path correctly and establish a deployment procedure.
One way to make the transition from development to production a smooth process is to use the same directory for your files. If you are using one Drupal installation for each project sites/default/files will work fine. If you set your file system path specific to your development hostname (for example: site/dev.HOSTNAME.com/files), you'll be in trouble when you transition to a new hostname.
Here's an example process to show how you could transition a site from development to test (or any other environment).
1. Migrate your MySQL database to the new server. Each environment should use its own database.
# connect to mysql as a privileged user to create a new database and user
$ mysql -u USER -pPASSWORD -h NEWSERVERHOST
> create database NEWDATABASE;
> GRANT ALL PRIVILEGES ON NEWDATABASE.* TO 'NEWUSER'@'localhost' IDENTIFIED BY 'NEWPASSWORD';
# backup database
$ mysqldump -u USER -pPASSWORD DATABASE > DATABASE.sql
# import database
$ mysql -u NEWUSER -pNEWPASSWORD -h NEWSERVERHOST NEWDATABASE < DATABASE.sql
2. Copy your Drupal installation to the new server. Hopefully, you're using subversion (or some other version control software) to deploy these files.
3. Create a new folder in your sites directory for the new hostname. Copy your existing settings.php into the new folder. Edit this file and update the MySQL connection string to match your new database, user, and password.
4. Create a new Apache virtual host for your new hostname and direct DNS accordingly.
Now, if DNS has been setup for your new hostname, the site should be up and running on the new server with a new hostname.
Additional thoughts: It's very important to avoid hard coding hostnames and absolute paths in your links, code, and themes. I promise, it will make your deployment process a nightmare.