You have two options - to setup manually or to use One-Click Apps.
For first you need to setup your Droplet or i.e. VPS. For OS i recommend you Ubuntu (16.04 x64) as most popular distro and one with most tutorials and community support.

It's advised that you follow Initial Setup guide that will help you setup basic things.

Now, you need Web server, a thing that will serve your webpages. Don't forget PHP and/or MySQL. All this you can get in LAMP stack or LEMP stack. Difference between this two are in Web server. LAMP comes with Apache, while LEMP with Nginx.
On basic level, there are no key difference between it, nginx could be less memory intensive. Apache is more popular, more used and could have better community support but this is discutable.

Anyways you need to follow tutorial for one of it - How To Install Linux, Apache, MySQL, PHP (LAMP) stack on Ubuntu 16.04 or How To Install Linux, Nginx, MySQL, PHP (LEMP stack) in Ubuntu 16.04.

This steps could be skipped by using One-Click App. You can choose LAMP One-Click App or LEMP One-Click App.

Now you need to copy all files from Hostgator to DigitalOcean. This is step for both solutions, so you need to follow it. To upload files to DigitalOcean you can use scp from your terminal. If you use Windows, WinSCP will help you, it even has GUI.
You need to download files to your machine from Hpstgator (probably using FTP, check help for it). When you download use scp to copy to DigitalOcean in /var/www/html.
You can setup FTP and use it instead of scp as alternative solution.

Don't forget database if you use one. :)
Check now does you see fully working site on Droplet IP.

If yes, go to domain settings and change A record to Droplet IP. It could need some time to refresh DNS but after it, it will point to DigitalOcean with zero downtime. :)

Feel free to ask me anything, I or somebody will try to help you. ;)