When using WordPress you have the benefit of their huge library of community created plugins and addons. The easiest way to accomplish this migration would be to use a WordPress plugin that allows you to export and import your site directly from wp-admin (including files and database). If you choose this option you can create a new droplet using our WordPress One-click image and then install the plugin on both instances and migrate all your data.
That will take care of the actual data. For the domain name you can do the following:
1.) [Optional but Recommended] Test your migration: First, we will test to make sure everything on the new server works as expected. You can do this by creating a hosts file entry. This method will have your computer use a specific IP address for the domain name rather than using what is provided by the public DNS.
This guide will show you how to create a hosts file entry locally. Create one pointing the domain name you are using for your WordPress site and your new droplet’s IP address. Load your site using the domain name in a browser, the version you see should now be the version on your new server that is not yet publicly available on your domain. If everything looks good, delete the hosts file entry. Leaving the entry in place could cause issues or at least confusion for you in the future.
2.) Create a new DNS zone on DigitalOcean: This guide will walk you through creating a new DNS record for your domain on the DigitalOcean nameservers.
3.) Update your DNS nameservers to ns1, ns2 and ns3.digitalocean.com
Once this is done we would recommend allowing 12-24 hours before deleting or otherwise taking down the site on GoDaddy if possible. This will ensure that anyone with DNS results that were cached on their end before the change are still able to view the site. If your site allows commenting or other user login functions you may wish to disable them in the old copy of the site during this time to make sure nobody adds new content to the old server.
If you’d prefer to migrate your files and database manually, let me know and I can provide a step by step guide for that option as well.