@CorryDye-2623 Are you planning to use the Azure data science VM offering from Azure as a service which contains pre-installed tools and packages that can be deployed on windows 10/Ubuntu or are you looking to setup a VM on Azure and then install your own packages and maintain a VM as a data science virtual machine?
If you are opting for either of the two then the process to deploy a VM in a VNET are pretty much the same except with a DSVM the NIC security group is pre-configured where certain ports are open based on the tools that are installed. If you plan to deploy a standalone windows 10 VM then you can customize the security groups to the required ports. A DSVM does not support accelerated networking so latency could be higher.
In both the cases you can also use Azure AD to manage access and login to your machines to provide secure access to required users.
Also, you cannot use the local desktop to reduce cost of your Azure VM. To reduce cost you could choose spot VMs which are cheaper than pay-as-you-go, you should ensure your applications are designed to handle interruptions and don't run production workloads to take advantage of the cost savings.
With respect to connecting surface to the VM, I am not too familiar with this experience. I hope the community could chip in with their experience. Thanks!!