Step number one complete. I've got my instance (called a Droplot) started on the DigitalOcean service. Their UX is vastly superior to any other VM reseller I've seen before. I picked the image they have that comes pre-installed with Node, and quickly had the Ghost blogging engine running.
Next step is to change the DNS servers so that
www.stevebrownlee.com will resolve to the new server. Then I'll get Mailgun set up, and that should be it.