If you’re wondering if there is a cost to port, then I think you misunderstand how this hosting works.
Digital Ocean provides, for a fee, Linux servers. Ones that you set up. They won’t come with Apache/NGINX, PHP/HHVM, or MariaDB/MySQL/Percona. You will be responsible for installing all this yourself.
Basically, you get a blank slate server, with SSH access. What you do from there is up to you. There are plenty of docs on how to get going in DO’s Tutorials section, but again, it’s on you to build out your server to fit your needs.
There are guides on transferring your data, but you’ll need to download everything from HostGator first.
Another point worth mentioning: Digital Ocean doesn’t provide Cpanel or any other web admin panel for you to manage your server. Just a place to build, destroy, and back up servers. You will also have to learn how to set up a mail server, or learn how to use Google to handle your site’s mail.
This is very much a self-service hosting provider, geared towards admins and developers.