Just follow the standard process for going live. Copy all the files and the database to a new server, updating paths etc.
When it’s live on the new domain, you should get an email from modbot telling you there’s a license issue on the new domain. (You can open the SimpleCart component to speed up when your receive the email). There’s a link in that email to the site, where you will be asked to assign a new license, or to transfer the license. As you’re using a free dev license currently, only the assign new license option will be available.
At that point it can either use a license you already purchased, or you can purchase a license then and use that. After clicking Assign License, the site will automatically download the new license key for the domain.