I will try to give you some idea how it can be done.
1. Control panel
CPanel is most used control panel by web hosting providers. Main negative downside is it's price. As far as I see on their web site, you need to $200 yearly for it. Let's agree, it's not a small price.
On other side ZPanel is free alternative. But as it looks... it is not maintained project anymore. Their latest commit (change) on GitHub was in 2014!
I would consider you to research other alternatives. What comes to my mind is VestaCP and Ajentis CP with AjentiV.
DigitalOcean have tutorials for both of these.
DigitalOcean - How to install VestaCP on Ubuntu 16.04
Using the Ajenti Control Panel with Ajenti V
If you question about DigitalOcean platform and more details about migration you can contact sales team here.
First of all you will need to create Droplet (this is VPS) and do initial config. There is pretty good tutorial how to do it.
Size of droplet is something that you should decide, it depends on size of disk you need and how much sites you have, and what is their job.
About OS, I would personally recommend Ubuntu (16.04) because most of tutorials and documentation here is made on base of it.
After that install all software you need. You will need LAMP (Linux-Apache-Mysql-Php) or LEMP (Linux-Nginx-Mysql-Php) stack. My experience is that most web hosting providers use Apache for webserver so you could go with LAMP.
Than... you need to backup old server and transfer to new server.
That include all files from server.
All databases for server (you probably have phpmyadmin on current server, so you can use that).
You can save all databases to your computer, copy to new server, execute .sql files.
About files, copy all to your computer, and transfer to new server on appropriate.
Keep in mind one big thing: Don't let users change any data after backup is made and until website is up on new server. If user change any data on old server, it will be lost!
After you setup and test server you need to point all domains to new server.
This is some short explanation on how migration can be done.
If you need more help, ask here or don't hesitate to fire up a ticket to support. DigitalOcean have epic support, and they can help about anything :D