To do it your self, you may follow the below steps:
1. To transfer public_html contents, generate a zip file of the public_html from a remote host using the cPanel file manager.
2. Then download it to your local machine.
3. Upload it to your account with us via FTP or File Manager (if small in size)
4. Then extract it to the appropriate location via File manager.
For email data, you may zip /home/username/email/ directory like this and restore it to the corresponding location here. Also copy /home/username/etc directory as well which contains email user and password information.
For copying databases over, you may generate .sql dump of the database via phpMyAdmin and download it. Then import it to the corresponding database via phpMyadmin here or using cPanel >>> Backup >>> Restore Mysql Database backup.