It's really not that difficult, but you'll need to inform the new clients of a few issues they'll run into with the change.
Assuming you'll be transferring everything using the Cpanel move features, that'll be your first step. Once everything has been transferred, it probably wouldn't hurt to synch up the data with Rsync over SSH. Have a new copy of your Apache configuration ready on the server clients are being transferred from with Redirects to the new server so you don't end up with database problems. Once you've finish the synch process, move this Apache backup config into place and restart Apache.
Since you'll have control of the old host's domains I'm assuming, you'll be able to re-register the nameservers with IPs on the new server so you don't have to tell clients to change their nameserver information with their registrars.
The only problem left is with email. You'll need to let clients know that for the next few days, they'll need to check email on both servers since I don't believe you'll find an easy way to have that all redirected to the new server. Somebody who's done these types of moves alot may have an option for you.
I hope that helps somewhat and didn't confuse you even more.