For internal server to server communications in Azure, what certificates are best to be used other than the self signed certificates?
I don't think it's a matter of "best", as it depends on your orgnization's resources and needs.
Considering it's only for internal communication, if your organization has a Certificate Authority as part of their IT infrastructure, you might be able to generate a certificate for your needs.
If your organization doesn't have one, the other option is buying a certificate from a trusted issuer, but I personally feel it's an overkill for internal communication.
For internal communication, if there are no transport security requirements defined (such as by regulations), the default HTTPS is OK (if your using Azure App Service for example).
Hope it helps!