Q. If I convert an Azure VM D/G series to a DS/GS series will the storage be migrated to Premium storage?
A. It is possible to resize Azure VMs and even change the SKUs using the Azure portal or even PowerShell, for example:
$vm.HardwareProfile.VirtualMachineSize = "<new size>" | Update-AzureVM
Remember if you change the size or SKU of a VM you will lose the content of the temporary drive (however you should never consider the temporary drive as persistent and never store data you care about on the temporary drive). Remember that VMs are deployed to cloud services that is bound to a cluster in Azure and the SKU/size can only be changed to a SKU/size supported in that cluster when using Azure Service Manager. In FAQ http://windowsitpro.com/azure/virtual-machine-sizeseries-limitations-when-addingresizing-cloud-service I walk through these types of clusters. When changing the VM size/SKU you will be limited to those supported within the cluster. If the VM is running on Azure Resource Manager and is deallocated you can change the SKU/size to any available in the region because ARM VMs are not bound to cloud services and therefore not bound to a cluster.
If you change a VM from a D/G to a DS/GS the storage will not be moved from Azure Storage to Azure Premium Storage. It will remain on Azure Storage. If you wish to migrate to Azure Premium Storage you would need to delete the VM while keeping storage, move the disks to Azure Premium Storage then create a new DS/GS series VM using the copied disks.
Microsoft has a great detailed article at http://blogs.msdn.com/b/igorpag/archive/2015/11/04/azure-vm-resizing-and-sku-change-in-asm-and-arm.aspx which walks through additional detail.