How to move a site from one cPanel host to another

Connect through FTP to your old hosting address and download all the necessary HTML files from the public_html folder.
Connect through cpanel to your old hosting address and download all the necessary databases. For a wordpress you have to download one database only. To do so go to Files / Backups and click on the desired database to download it. If you already have a wordpress site, install updraft plus plugin and create the backup (it will generate a gz file which is much smaller).
Transfer the email accounts and the associated mails. To do view the directions here.
Connect through cpanel to your new hosting address and go to Databases / MySQL Databases. Create a new database, a new user and add the user to the database (using all the privileges).
Go to Databases / PhpMyAdmin, choose your database from step 4 and import the database from step 2.
If you move a wordpress site, open wp-config.php and change the database name, username and password, so that the wordpress points to the new database and user.
Connect through FTP to your new hosting address and upload the HTML files from step 1 to the public_html folder. If your site does not have a domain name yet and only exists in your hosting space, use the IP of the new site.
Test your new site to see that it actually works. To do so, see this tutorial.
Finally, go to to the administration panel of your domain name and change the DNS name servers.
Wait for 4 to 48 hours for the propagation to complete.
Check that the site is working.
Check that the emails are working.