DO is un managed so you have to get everything done by your self..
I too started like you.. Moved from a cPanel setup to my own server here..
Create your server..
In my case I chose CentOS 6.5 x86.. Since I wanted the 512MB plan so choose 32bit OS..
Back up everything in your old Host
In my case, I have three domains, with WordPress and CopperMine so I zipped the website files and downloaded the database from phpmyadmin..
Configure the basics of OS Setup..
In my case it was CentOS 6.5 hence I followed that tutorial available here.. If you choose anything else, follow that tutorial..
Install the needed services..
Typically that would be Apache / Bind / mail servers and such.. SInce I was a newbie and coming from cPanel based hosting, so I tried few Control Panel, I decided to give zPanel a try..
So I installed zPanel (Now Sentora Panel), which comes with everything that web server would need..
There are many free panels, you can try and choose the one you like..
if you mess up, no issues, just rebuild the droplet from scratch and try something else..
Once you get things installed properly now you can move towards transferring the website..
Update your domain DNS at registrants end, then setup DO DNS..
Add the domain in your control panel acc
This is in your hosting control panel, like zPanel of mine, you can have your own panel, there add the domain, which will create its file system and DNS.. Edit DNS entry in panel if required..
Use your panel (in my case zPanel) to create emails / forward / ftp or what ever you need..
Start the transfer
Upload the web content in the designated directory of your new server.. Create and Map the database as per new settings..
Wait for DNS to update
Usually takes 6 to 8 hours for DNS to update world wide..
If you find everything running fine, start doing few things like Firewall config and swap file and things which might be related to fine tune your setup..