Q. What is the best way to routinely copy large amounts of data to Azure?
A. Azure has an import/export service that allows data to be sent to Azure via physical media however there is a latency due to the shipping of the media and time to import. Another option would be to combine a high bandwidth ExpressRoute connection (dedicated connection from on-premises to Azure that goes up to 10 Gbps) and the AzCopy utility which copies data to/from Azure storage accounts.
AzCopy can be downloaded from https://aka.ms/downloadazcopy. To use AzCopy the storage account key is required which can be found via the Access keys tab of the storage account in the portal or using PowerShell (http://windowsitpro.com/azure/get-storage-keys-using-powershell-azurerm or http://windowsitpro.com/azure/use-powershell-obtain-azure-storage-account-key).
The syntax is simple (the /s means a recursive copy including all sub-folders):
Azcopy /source:<source folder> /dest:<URL of the storage account> /destkey:<storage account key> /s