From your explanation about the server you currently use, you’ll need mainly a LAMP stack server and an Email service - Postfix or Exim.
Let’s start with the LAMP stack, this will include - Apache, MySQL, PHP. I can see you’ve marked CentOS as an OS so I’ll recommend you to check this article how to configure LAMP on a DigitalOcean droplet using CentOS 7 - HERE
I’ll recommend using Postfix for your setup, it’s pretty straight forward to configure and to use.
Good thing is that there is a good article in the DigitalOcean knowledge base again - HERE
The above two should be enough for you to get started and move your websites over.
As for the actual migration of the websites, I’ll recommend you to first configure your Web Server and Email service and then perform the migration.
This tutorial explains how to install postfix and cyrus with yum, how to configure postfix, how to configure cyrus, how to install mailx, and how to send a test email. Postfix is free open source Mail Transfer Agent which works to route and deliver email.This tutorial should help you set up a mail server on a centos droplet.