A Plesk-to-cPanel migration is perhaps less painful than some. Here are some basic steps to help you (this process requires a moderate level of administrative knowledge:)
If you don't have root access, you will migrate everything via FTP or SCP. First, create the account on your new cPanel server.
1) You can either directly FTP from your cPanel server to the Plesk account OR download the data from the Plesk server on your local machine and upload to public_html directory. Make sure you change the ownership of the files/directories to username.username on the cPanel server using cPanel's File Manager, or the "chown" command via SSH.
2) Create the databases/db_users from cPanel's Databases area. Make a dump of the databases on the Plesk server via phpMyAdmin and restore them into the new databases. cPanel always attaches the account username to the database and db user. You will have to make the appropriate changes in any website config files, such as config.php.
3) Emails cannot be migrated from Plesk, so download the emails from the Plesk server onto your local machine in Outlook, Thunderbird or your favorite email client so you can have them stored locally.
4) Once everything is migrated and tested, you can make any required DNS changes.