Here are the steps to take:
1) Move the entire Director folder to the new server. Make sure permissions are retained for the following content folders:
album-audio
albums
app/tmp
xml_cache
2) Delete any folders/files inside the following folders:
xml_cache
app/tmp/cache
3) Export your database at your old host and import it into the database at the new host. Consult the technical support for the respective hosts for more info on how to do this.
4) Change the config/conf.php file to reflect the connection details to your new database.
5) Attempt to sign-in to Director. You do not need to reinstall. If all goes well, you should be able to login and use Director as you did at your old host.