This is not my area of expertise, and as I mentioned, we are limited in the support we can offer via the user forum on these topics. However, none of this information is specific to AtoM, so you can search online for more guides and resources to help you. I will try to provide a few thoughts below.
How you go about this will depend on the service provider. In many cases, they will provide you with a virtual server, and you will need to reinstall AtoM on their infrastructure. So: you would review their offerings, select a hosting plan that includes an Ubuntu VM with command-line access, and then follow their access documentation. Once you have access, you would install AtoM there. In other cases they may have tools to help migrate local content, or their services may include assistance with installation and configuration - it depends on what you choose and pay for.
The hosting provider may or may not also have domain name services - it depends on the vendor and the service, but if not, then you would separately contact a Domain Registrar, choose a domain name that is available, pay for it, and provide them with the public IP address and port of your AtoM install on the hosted platform. They will configure the DNS record and it will start working within 1-2 days, usually.
Otherwise, you can try contacting a general web hosting provider like Digital Ocean directly to ask more questions about their service offerings. Good luck!