I have done this several times as follows:
1. Backup the existing moodle file (tar cvf moodle.tar moodle)
3. Backup the moodledata file ( tar cfv moodledata.tar moodledata)
1. FTP moodle.tar to the web home directory and (tar xvf moodle.tar)
2. Create a moodle database in MYSQL and restore the backup database.
3. FTP moodledata to your home directory and (tar xvf moodledata.tar )
4. Go to the config.php file and update with the information of the new server.
And everything should work OK providing your new server meets the requirements as stated in the install documentation.