If you need to migrate large amount of data from another service provider, we can certainly help you. (For small amount of data, you can easily download your files from the other service provider and then upload them to DriveHQ). Please contact DriveHQ customer support for more details.
For initial data backup / upload, if you have large amount of data (e.g. if you have more than a few hundred GBs of data), you can copy your files to an external drive and send the drive to us. We can help you upload from our data center. This way, it is much faster and it will not boggle down your local network. Once the initial upload has finished, the subsequent uploads will be very fast as it will only upload the new or changed files.
FYI, from DriveHQ.com website, you can use the "New File" feature to download files from another website directly to your DriveHQ storage. The feature is offered as a convenient way to grab some files from the Internet; it may not be robust enough to grab very large files.