@Richard Dunning Thank you for your interest in Azure.
I personally deal with Azure Web Apps, which will make my response a little bias but I can hopefully help address your questions and provide you further documentation.
In regards to hosting your app, I would suggest using Azure Web Apps. This platform is great because it reduces the overhead management of applying patches and general maintenance. Here is a an overview of Azure Web Apps and here is a quick start for setting up a C# ASP.NET Core app on an Azure Web App. Web Apps are great because they also include much of what you need to get started, such as authentication support.
In regards to the DB, there are few routes you can go. If you wish to keep your current DB and run it on-prem, you can use an Azure Web App feature called hybrid connection, which allows you to install an agent on your DB machine and the agent allows the DB and web app to communicate securely, reducing the work required for a migration to Azure and allowing you to maintain on-prem control of your DB.
If you're looking to move away from on-prem server management and fully leverage the cloud, we do have a Azure MySQL DB offering and documentation on how to connect to your web app.
If you were to ask your question over in Azure-VM, they might share a different approach but I believe that your scenario would be a great fit for a PaaS type solution that is outlined above because it reduces the amount of management and input required from you to make it work.
Once you have reviewed the documentation, please let me know if you have any further questions. I also believe I can provide you with a link to contact our sales team if you're wanting more of a 1:1 interaction.