There are many simple ways of doing it,
1.first on your old server go to backup in cpanel
2.backup your data
3.make sure you select the ftp destination when you are asked
4.enter your ftp details of the new server and pree ok.
5.wait for some time and ur data will be transfered to the new server.
this is only posible if U have access to SSH(Shell)
1.backup your data in the old server
2.after backing up,set permission of the backed up file to 777
3.login into ssh of new server
4.if you are in root folder,then navigate to ur accunt using :
Where the account username is your cpanel usernameCode:cd /home/account_username
4.copy data into new server using the shell command
5.wait for sometime and ur file is tranferedCode:wget http://ur_site_url/backup_file_name.tar.gz
if U own your own server,then login to your whm
go to copy website from another server,enter old cpanel details and wait for it to copy.
I will upload a script to do that here soon