This is the obligatory “If you can read this, the move was successful” message. Moving to the new server was very easy, I only had to give my cPanel details to the support staff, and ten minutes later they had my site and e-mail up and running.
There was only one Perl module missing to get my applications going, which was fixed in another two minutes. Now I have to wait for the new DNS settings to propagate. This was a lot easier than I thought!